Organise the long search on which one's suit may depend (7,6)
I believe the answer is:
clothes hanger
'on which one's suit may depend' is the definition.
(clothes hang or depend on a hanger)
'organise the long search' is the wordplay.
'organise' indicates an anagram.
'the'+'long'+'search'='thelongsearch'
'thelongsearch' anagrammed gives 'CLOTHES HANGER'.
(Other definitions for clothes hanger that I've seen before include "Wardrobe kit" , "one's hooked" .)