I think the answer is
When it's fat then it's thin.
"Fat chance" refers to something that is unlikely to happen (the chance is actually small).
When it's slim then there's min.
"Slim chance" refers to a very small possibility (a minimum chance)
When it's fights there is hope.
A "fighting chance" refers to a situation where there is a small possibility of success with a lot of hope
'Til it's outside, then nope.
"Outside chance" refers to a remote possibility.
You can take it if you think you must.
"Take a chance" refers to the act of doing something that has a questionable probability of success
You can give it if you're feeling trust.
"Give a chance" means to allow or grant someone or something the opportunity to do something, usually requiring a degree of trust.
The first one's for whenever you're able.
"First chance" refers to the first opportunity to do something.
The last one, you might find a poker table.
"Last chance" refers to the last hope of success which would be appropriate at a poker table when one goes all in.
OP's comment: This line refers to the "last chance saloon" - a difficult situation in which there is one final chance to put it right (the saloon putting us in mind of card games in the Old West).
"On the off-chance" means "just in case".