Hired escort in carriage, unaccompanied, heading off (6)
I believe the answer is:
gigolo
'hired escort' is the definition.
'carriage unaccompanied heading off' is the wordplay.
'carriage' becomes 'gig' (gig is a kind of carriage).
'unaccompanied' becomes 'solo' (I've seen this before).
'heading off' means to remove the first letter.
'solo' with its first letter taken away is 'olo'.
'gig'+'olo'='GIGOLO'
'in' is the link.
(Other definitions for gigolo that I've seen before include "Hired male escort" , "Old word for man maintained by rich woman" , "dancing partner" , "Lounge lizard" , "Kept man" .)