Avant-garde tailors carrying on referring to hairdressers (9)
I believe the answer is:
tonsorial
'referring to hairdressers' is the definition.
(tonsorial can mean relating to hairdressing)
'avant-garde tailors carrying on' is the wordplay.
'avant-garde' is an anagram indicator (a new or avant-garde version of the letters).
'carrying' indicates putting letters inside.
'tailors' is an anagram of 'tsorial'.
'tsorial' placed around 'on' is 'TONSORIAL'.
(Other definitions for tonsorial that I've seen before include "for barber" , "using scissors for work?" , "Barber's" .)