Jazz trumpeter with stomach upset (7)

I believe the answer is:

satchmo

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'jazz trumpeter' is the definition.
'satchmo' can be an answer for 'trumpeter' (I've seen this before). I'm unsure of the 'jazz' bit.

'stomach upset' is the wordplay.
'upset' indicates an anagram.
'stomach' anagrammed gives 'SATCHMO'.

'with' is the link.

Can you help me to learn more?

(Other definitions for satchmo that I've seen before include "Louis Armstrong's nickname" , "trumpeter" , "Louis Armstrong's byname" , "Name used by jazz player Louis Armstrong" , "Jazz musician" .)

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