The most common and most likely answer to this crossword clue is the 9 plus 5 letter word SPAGHETTI STRAP
I've seen this crossword clue in The Sunday Telegraph
S
P
A
G
H
E
T
T
I
S
T
R
A
P
Buddy explains!
The straight part
We know that "Supporting part of garment" is the straight/definition part of the clue. The straight part is often a synonym or definition, in this case supporting part of garment = spaghetti strap.The cryptic/wordplay part
In cryptic crossword there's often an wordplay/cryptic part as well.This part involves a more complex hint, such as an anagram, homophone, hidden word, etc.
Sadly, We don't have an explanation for this specific crossword clue yet.