Scrap, initially overlooked, vet gives for recycling (7)
I believe the answer is:
'scrap' is the definition.
'initially overlooked vet gives for recycling' is the wordplay.
'initially overlooked' means to remove the first letter.
'for recycling' is an anagram indicator.
'vetgives' with its first letter removed is 'etgives'.
'etgives' with letters rearranged gives 'VESTIGE'.
(Other definitions for vestige that I've seen before include "2 [HINT]" , "Small trace" , "'Shadow, trace (7)'" , "Slight amount, hint" , "(Last) trace" .)