Shreds every bit of the wool fabric (10)

I believe the answer is:

tattersall

I'm a little stuck... Click here to teach me more about this clue!

'bit of the wool fabric' is the definition.
'tattersall' can be an answer for 'fabric' (I've seen this before). I'm not certain of the 'bit of the wool' bit.

'shreds every' is the wordplay.
'shreds' becomes 'tatters' ('tatter' can be a synonym of 'shred').
'every' becomes 'all' (similar in meaning).
'tatters'+'all'='TATTERSALL'

Can you help me to learn more?

(Another definition for tattersall that I've seen is " fabric".)

I've seen this clue in the Sydney Morning Herald.

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