The magic hat word series

Question

A magician pulls one thing after another out of his hat. How could that little hat contain so many things?!

We're going to do something similar, by pulling the entire alphabet out of a word.

Start with a word.
The word should contain (at least one) "A". Pull the "A" out of the word, show it to the audience with dramatic flourish, and toss it away. Quietly replace it with a new letter of your choosing. Rearrange to form a new word.
If you've played it correctly, the word at this point should contain (at least one) "B". Pull the "B" out of the word, show it to the audience with dramatic flourish, and toss it away. Quietly replace it with a new letter of your choosing. Rearrange to form a new word.
If you've played it correctly, the word at this point should contain (at least one) "C". Pull the "C" out of the word, show it to the audience with dramatic flourish, and toss it away. Quietly replace it with a new letter of your choosing. Rearrange to form a new word.
If you've played it correctly, the word at this point should contain (at least one) "D". Pull the "D" out of the word, show it to the audience with dramatic flourish, and toss it away. Quietly replace it with a new letter of your choosing. Rearrange to form a new word.
If you've played it correctly, the word at this point should contain (at least one) "E". Pull the "E" out of the word, show it to the audience with dramatic flourish, and toss it away. Quietly replace it with a new letter of your choosing. Rearrange to form a new word.
If you've played it correctly, the word at this point should contain (at least one) "F". Pull the "F" out of the word, show it to the audience with dramatic flourish, and toss it away. Quietly replace it with a new letter of your choosing. Rearrange to form a new word.
If you've played it correctly, the word at this point should contain (at least one) "G". Pull the "G" out of the word, show it to the audience with dramatic flourish, and toss it away. Quietly replace it with a new letter of your choosing. Rearrange to form a new word.
If you've played it correctly, the word at this point should contain (at least one) "H". Pull the "H" out of the word, show it to the audience with dramatic flourish, and toss it away. Quietly replace it with a new letter of your choosing. Rearrange to form a new word.

The audience is going wild! Give the illusion of picking up the pace in the next step:
If you've played it correctly, the word at this point should contain both an "I" and a "J". Pull both the "I" and the "J" out of the word at the same time, show them to the audience with dramatic flourish, and toss them away. Quietly replace them with TWO new letters of your choosing. Rearrange to form a new word.

If you've played it correctly, the word at this point should contain (at least one) "K". Pull the "K" out of the word, show it to the audience with dramatic flourish, and toss it away. Quietly replace it with a new letter of your choosing. Rearrange to form a new word.
If you've played it correctly, the word at this point should contain (at least one) "L". Pull the "L" out of the word, show it to the audience with dramatic flourish, and toss it away. Quietly replace it with a new letter of your choosing. Rearrange to form a new word.
If you've played it correctly, the word at this point should contain (at least one) "M". Pull the "M" out of the word, show it to the audience with dramatic flourish, and toss it away. Quietly replace it with a new letter of your choosing. Rearrange to form a new word.
If you've played it correctly, the word at this point should contain (at least one) "N". Pull the "N" out of the word, show it to the audience with dramatic flourish, and toss it away. Quietly replace it with a new letter of your choosing. Rearrange to form a new word.
If you've played it correctly, the word at this point should contain (at least one) "O". Pull the "O" out of the word, show it to the audience with dramatic flourish, and toss it away. Quietly replace it with a new letter of your choosing. Rearrange to form a new word.

Careful now, you are going to need some additional sleight-of-hand after the "P" in order to accommodate the tricky "Q":
If you've played it correctly, the word at this point should contain (at least one) "P". Pull the "P" out of the word, show it to the audience with dramatic flourish, and toss it away. Quietly replace it with TWO NEW LETTERS. No one will notice that the word got a little longer! Rearrange to form a new word.
If you've played it correctly, the word at this point should contain (at least one) "Q". Pull the "Q" out of the word, show it to the audience with dramatic flourish, and toss it away. Meanwhile, with your other hand, discreetly remove the second letter from the word, palm it and put it in your pocket. No one needs to know about it! Quietly add ONE new letter of your choosing. You're back to the original word length. Rearrange to form a new word.

If you've played it correctly, the word at this point should contain (at least one) "R". Pull the "R" out of the word, show it to the audience with dramatic flourish, and toss it away. Quietly replace it with a new letter of your choosing. Rearrange to form a new word.
If you've played it correctly, the word at this point should contain (at least one) "S". Pull the "S" out of the word, show it to the audience with dramatic flourish, and toss it away. Quietly replace it with a new letter of your choosing. Rearrange to form a new word.
If you've played it correctly, the word at this point should contain (at least one) "T". Pull the "T" out of the word, show it to the audience with dramatic flourish, and toss it away. Quietly replace it with a new letter of your choosing. Rearrange to form a new word.
If you've played it correctly, the word at this point should contain (at least one) "U". Pull the "U" out of the word, show it to the audience with dramatic flourish, and toss it away. Quietly replace it with a new letter of your choosing. Rearrange to form a new word.
If you've played it correctly, the word at this point should contain (at least one) "V". Pull the "V" out of the word, show it to the audience with dramatic flourish, and toss it away. Quietly replace it with a new letter of your choosing. Rearrange to form a new word.
If you've played it correctly, the word at this point should contain (at least one) "W". Pull the "W" out of the word, show it to the audience with dramatic flourish, and toss it away. Quietly replace it with a new letter of your choosing. Rearrange to form a new word.
If you've played it correctly, the word at this point should contain (at least one) "X". Pull the "X" out of the word, show it to the audience with dramatic flourish, and toss it away. Quietly replace it with a new letter of your choosing. Rearrange to form a new word.
If you've played it correctly, the word at this point should contain (at least one) "Y". Pull the "Y" out of the word, show it to the audience with dramatic flourish, and toss it away. Quietly replace it with a new letter of your choosing. Rearrange to form a new word.
If you've played it correctly, the word at this point should contain (at least one) "Z". Pull the "Z" out of the word, show it to the audience with dramatic flourish, and toss it away. Quietly replace it with a new letter of your choosing. Rearrange to form a new word.

And that's the whole alphabet!

(Applause)

Whew, take a bow.

Here's a dud example, just to illustrate how it might look:

SARDINE
        (pull out "A" and replace with "B")
BINDERS
        (pull out "B" and replace with "C")
DISCERN
        (pull out "C" and replace with "F")
FRIENDS
        (pull out "D" and replace with "E")
REFINES
        (pull out "E" and replace with "G")
FINGERS
        (pull out "F" and replace with "S")
SINGERS
        (pull out "G" and replace with "U")
SUNRISE
        (...)

This example runs out of fuel before it gets through the alphabet, but you get the idea.

For everyone's enjoyment, please try to favor more well-known words in your solution.

---

The series given by @Gareth McCaughan are correct. Here is another solution:

ABIDE
(pull out "A" and replace with "T")
DEBIT
(pull out "B" and replace with "C")
CITED
(pull out "C" and replace with "F")
FETID
(pull out "D" and replace with "H")
THIEF
(pull out "E" and replace with "G")
FIGHT
(pull out "F" and replace with "S")
SIGHT
(pull out "G" and replace with "L")
HILTS
(pull out "H" and replace with "J")
JILTS
(pull out "I" and "J" and replace with "A" and "K")
TALKS
(pull out "K" and replace with "M")
MALTS
(pull out "L" and replace with "R")
SMART
(pull out "M" and replace with "N")
RANTS
(pull out "N" and replace with "O")
ROAST
(pull out "O" and replace with "P")
PARTS
(pull out "P" and replace with "Q" and "U")
QUARTS
(pull out "Q" and "U" and replace with "E")
RATES
(pull out "R" and replace with "S")
SEATS
(pull out "S" and replace with "V")
STAVE
(pull out "T" and replace with "U")
SUAVE
(pull out "U" and replace with "W")
WAVES
(pull out "V" and replace with "X")
WAXES
(pull out "W" and replace with "M")
EXAMS
(pull out "X" and replace with "Y")
SEAMY
(pull out "Y" and replace with "Z")
MAZES
(pull out "Z" and replace with "U")
AMUSE

If you want to try constructing a solution for yourself, these are good starting words:

ABIDE
BAITS
BASED
BATED
BEAST
IDEAS

Answer 1

Computer-assisted answer

This seems to me like it's awfully difficult without computerized assistance, so I didn't try it that way. I made a couple of assumptions: (1) when we replace a letter with "a new letter", it's OK for the new letter to be the same as the old; (2) a "word" is anything in some appropriate word list that contains only lowercase letters. (No capitals, likely to denote proper nouns; no apostrophes or hyphens or anything.)

With a wordlist of about 74k "common" words I happened to have handy, I found no solutions. So unless my code is broken, any solution is going to have at least one somewhat obscure word in it. So I then used a larger dictionary, containing about 350k words, and found 882 candidate starting words -- but, looking at a smallish sample of the results, found that they all contained some really obscure words. So, Goldilocks-like, I switched to a dictionary of intermediate size, containing about 110k words. That yielded 133 starting words. Here's one fairly nice solution. I've emphasized the new letters in each word after the first.

sable -a-> bells -b-> cells -c-> dells -d-> fells -e-> fills -f-> gills -g-> hills -h-> jills -ij-> skull -k-> mulls -l-> mules -m-> lunes -n-> louse -o-> pules -p-> sequel -q-> rules -r-> lutes -s-> lutea -t-> value -u-> slave -v-> swale -w-> axles -x-> lyase -y-> lazes -z-> earls

Definitions of the most obscure words:

jill: a girl or woman, especially an ordinary one of low status
lune: a shape formed by two circular arcs
pule: to whine
lutea: yellow; only seems to occur in Latin-derived compounds, seems a bit of a cheat
swale: timber
lyase: an enzyme that catalyses bond-breaking by means other than hydrolysis and oxidation

After removing the (to my mind somewhat dodgy, though e.g. it is legal in Scrabble) lutea from the dictionary and re-running, here's a reasonably nice solution:

bales -a-> biles -b-> slice -c-> slide -d-> files -e-> flits -f-> gilts -g-> hilts -h-> jilts -ij-> stalk -k-> malts -l-> trams -m-> rants -n-> rotas -o-> strap -p-> quarts -q-> sutra -r-> talus -s-> talus -t-> ulvas -u-> slave -v-> swale -w-> axles -x-> lyase -y-> lazes -z-> earls

New obscure words:

sutra: a Buddhist sacred text
talus: the sloping side of a wall or earthwork in a fortification
ulva: a lettuce-like alga that grows in the sea

Those solutions make use of the ability to replace a letter with itself. OP has indicated that that wasn't the intention. The least obscure chain I've found (running my code, manually removing the nastiest words in the solutions from the dictionary, repeating, etc.) is:

abide bides dices sited heist shift sight hoist joist stoke stole tomes onset estop setup quests trues suets saute suave stave waste taxes yeast zetas baste

(I'd be happier without "estop" and "trues" but neither of them is all that bad.)