Insect heading off close to rug (6)
I believe the answer is:
earwig
'insect' is the definition.
(I know that earwig is a type of insect)
'heading off close to rug' is the wordplay.
'heading off' means to remove the first letter.
'close to' becomes 'near' (close to means nearby).
'rug' becomes 'wig' ('rug' is an informal term for a wig).
'near' with its first letter taken away is 'ear'.
'ear'+'wig'='EARWIG'
(Other definitions for earwig that I've seen before include "Aural kind of insect" , "It's a strange insect, I wager" , "Common garden insect" , "little insect" , "An insect; to eavesdrop (slang)" .)