Narrow part to cover with garment (9)
I believe the answer is:
waistcoat
'garment' is the definition.
(I know that waistcoat is a type of garment)
'narrow part to cover' is the wordplay.
'narrow part' becomes 'waist' (I've seen this before).
'to cover' becomes 'coat' (coating is a kind of covering).
'waist'+'coat'='WAISTCOAT'
'with' is the link.
(Other definitions for waistcoat that I've seen before include "Part of a three-piece suit" , "Clothing" , "Something worn" , "American's vest" , "A "vest" in the US" .)