Not wearing old hat demoralised (7)

I believe the answer is:

daunted

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

'not wearing old' is the wordplay.
'not' becomes 'un' (prefix meaning 'not').
'wearing' indicates putting letters inside.
'old' becomes 'dated' (dated can mean old or out of date).
'un' going inside 'dated' is 'DAUNTED'.

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(Other definitions for daunted that I've seen before include "Disheartened, being undated" , "Put off" , "Made discouraged, being undated" .)

I've seen this clue in The Daily Express.

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