Flight of American inside slammer (5)
I believe the answer is:
stair
'flight' is the definition.
(I've seen this before)
'american inside slammer' is the wordplay.
'american' becomes 'a' (common abbreviation - e.g. in organisation names).
'inside' means one lot of letters goes inside another.
'slammer' becomes 'stir' (I've seen this in another clue).
'a' put within 'stir' is 'STAIR'.
'of' is the link.
(Other definitions for stair that I've seen before include "Bit of a flight!" , "Word linking way and well" , "Rising step" , "One step in a series" , "Tarsi (anag)" .)