The most common and most likely answer to this crossword clue is the 7 plus 5 letter word CLOTHES HORSE
I've seen this crossword clue in Financial Times
C
L
O
T
H
E
S
H
O
R
S
E
More clues leading to the result CLOTHES HORSE
Buddy explains!
The straight part
We know that "Could you put your shirt on it?" is the straight/definition part of the clue. The straight part is often a synonym or definition, in this case could you put your shirt on it? = clothes horse.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.