Coat carried back, heading for exit (6)
I believe the answer is:
enrobe
'coat' is the definition.
(I know that enrobe is a more specific form of coat)
'carried back heading for exit' is the wordplay.
'carried' becomes 'borne' (past tense of 'bear' in the sense of 'carry').
'back' is a reversal indicator.
'heading for' suggests taking the first letters.
The initial letter of 'exit' is 'e'.
'borne' reversed gives 'enrob'.
'enrob'+'e'='ENROBE'
(Other definitions for enrobe that I've seen before include "make investment?" , "Clad ceremonially" , "Fancy" , "Don a vestment" , "put on formally ..." .)