Stainer’s daughter, the old composer’s last (4)
I believe the answer is:
dyer
'stainer's' is the definition.
(I've seen this before)
'daughter the old composer's last' is the wordplay.
'daughter' becomes 'd' (genealogical abbreviation).
'the old' becomes 'ye' (resembles a historical spelling of 'the').
'last' says to take the final letters.
The last letter of 'composer' is 'r'.
'd'+'ye'+'r'='DYER'
(Other definitions for dyer that I've seen before include "I change colour" , "Person who colours cloth" , "Person colouring cloth" , "Also a cleaner, probably" , "Cloth stainer" .)