Could it be a
You're very close
But I'm always ahead
When wearing a hat, you are very close to it, but it's always above ('ahead of'?) your head (so the word 'ahead' is sort of wordplay too).
Can be beheaded
but can't be fed
'Beheaded' is here used to mean 'removed from the head'. (The OP is not a native English speaker!) And certainly a hat can't be fed.
You can have
a look inside
there will be
no place to hide
try to enter me
more room arises
These verses are what makes it a
rather than just a
You can look inside this object, and indeed there appears to be no place to hide. But
when the conjuror is performing 'magic' with it on stage, he can produce all sorts of objects, most famously rabbits, from the apparently empty hat, and also make them disappear into it. So for a rabbit, trying to enter the hat may lead to 'more room arising suddenly'.
Credit to @d'alar'cop for the idea of a
but he couldn't make this fit with the 3rd verse. The only remaining mystery is the relevance of the multi-'had' sentence linked to by the OP - maybe it's a red herring, or just a hint to a word which sounds like 'had'?