Moment to fit the wig (6)
I believe the answer is:
weight
'moment' is the definition.
(thesaurus)
'fit the wig' is the wordplay.
'fit' is an anagram indicator (letters fitted or arranged in a different way).
'the'+'wig'='thewig'
'thewig' with letters rearranged gives 'WEIGHT'.
'to' is the link.
(Other definitions for weight that I've seen before include "The property of heaviness, measure of mass" , "Importance, heaviness" , "Gravitational pull" , "Heavy load - preponderance" , "gives a sense of oppression" .)