Rest of the pile (3)
I believe the answer is:
nap
I believe this clue is a double definition.
'rest' is the first definition.
(a nap is a short sleep)
'the pile' is the second definition.
(both can refer to the surface of a fabric)
'of' acts as a link.
(Other definitions for nap that I've seen before include "One likely to win" , "Sleep lightly" , "Short raised fibres on cloth on snooker table" , "Take forty winks" , "raised bits of velvet" .)