Scarf that's warmer, then cooler after removing half (7)

I believe the answer is:

muffler

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'scarf' is the definition.
(muffler is a kind of scarf)

'warmer then cooler after removing half' is the wordplay.
'warmer then' becomes 'muff' (I can't explain this - if you can you should believe this answer much more).
'after removing half' suggests halving the letters.
'cooler' with half the letters taken is 'ler'.
'muff'+'ler'='MUFFLER'

'that's' acts as a link.

Can you help me to learn more?

(Other definitions for muffler that I've seen before include "Sound deadener" , "Thick scarf - silencer (US)" , "Kind of scarf" , "Wrap or scarf worn round the neck" , "Warming garment" .)

I've seen this clue in The Independent.

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