By many an account, it's a short garment (3)
I believe the answer is:
mac
'garment' is the definition.
(mac is a kind of garment)
'by many an account it's a short' is the wordplay.
'by' is a charade indicator (letters next to each other).
'many' becomes 'm' (Roman numeral for a thousand).
'an account' becomes 'acc'.
'it's a short' means to remove the last letter.
'acc' with its final letter removed is 'ac'.
'm' put next to 'ac' is 'MAC'.
(Other definitions for mac that I've seen before include "PC" , "Coat - computer" , "A Gaelic son, often in name" , "one used in shower?" , "Overcoat" .)