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G'day Tdol,
May I suggest that the definition given is less than complete and may be confusing to the leanre.
At the drop of a hat is a fragmentary part of the more complete idiomatic saying, 'He'll blue (fight) at the drop of a hat.'

This does not mean that he will fight without hesitation but rather that he will fight with little or no provocation.
A dropped hat would have to be the most peacefully quiet slow motion that one could imagine of a bloke in the company of other blokes.
The falling hat makes no sound.
The falling hat travels quite slowly.
The falling hat breaks nothing.
The falling hat is not aggressive.

A bloke who will fight at the drop of a hat will fight with no provocation and on a whim. It is this aspect of whimsy that 'at the drop of a hat' conveys not a lack of hesitation.
I hope that I do not confuse your intent.

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