Saint on set crumpled headwear (7)

I believe the answer is:

stetson

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'headwear' is the definition.
I can't tell whether this definition defines the answer.

'saint on set crumpled' is the wordplay.
'saint' becomes 'st' (abbreviation).
'crumpled' is an anagram indicator (a crumbled or damaged version of the letters).
'on'+'set'='onset'
'onset' anagrammed gives 'etson'.
'st'+'etson'='STETSON'

Can you help me to learn more?

(Other definitions for stetson that I've seen before include "Hat for country-and-western singers, Texan oil magnates, etc." , "Felt hat" , "A wide-brimmed cowboy hat" , "St Neots (anag.)" , "Tip" .)

I've seen this clue in The Sun.

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