Supply yesterday's - first floor (6)
I believe the answer is:
storey
'floor' is the definition.
(US term for a floor in a building)
'supply yesterday's first' is the wordplay.
'supply' becomes 'store' (I've seen this in another clue).
'yesterday's first' becomes 'y' (1st letter of 'yesterday').
'store'+'y'='STOREY'
(Other definitions for storey that I've seen before include "part of the building" , "Oyster (anag) -- level" , "Layer of house" , "Floor or level in a building" , "rooms on one level" .)