Crossword Genius

Buffalo Bill going west for club (3)

Ross

I believe the answer is:

bat

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This clue might be a double definition.

'buffalo bill' is the first definition.
'bat' can be an answer for 'buffalo' (thesaurus). I'm unsure of the 'bill' bit.

'going west for club' is the second definition.
'bat' can be an answer for 'club' (as in sporting equipment). I am unsure of the 'going west for' bit.

Can you help me to learn more?

(Other definitions for bat that I've seen before include "once wielded by 60 [GREIG]" , "The cricketer wields it" , "Club used in cricket" , "Cricket, table-tennis and baseball implement" , "Small mammal or cricket striker" .)

I've seen this clue in The Guardian.
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