Satchmo's sort of music needs to speed up (4)
I believe the answer is:
trad
'satchmo's sort of music' is the definition.
(trad jazz - Louis Armstrong was nicknamed Satchmo)
'to speed up' is the wordplay.
'to speed' becomes 'dart' (darting is a kind of speeding).
'up' says the letters should be written backwards (in down clue: letters go upwards).
'dart' back-to-front is 'TRAD'.
'needs' acts as a link.
(Other definitions for trad that I've seen before include "Type of jazz or folk" , "(Of jazz) traditional" , "Classical (abbr)" , "Style of jazz" , "Conventional (jazz)" .)