Animal that's musical, by the sound of it (4)
I believe the answer is:
hare
'animal' is the definition.
(I have seen 'Swift animal ' mean 'hare' so perhaps 'animal' could also mean 'hare')
'musical by the sound of it' is the wordplay.
'musical' becomes 'hair' (1960s musical).
'by the sound of it' indicates a 'sounds like' (homophone) clue.
'hair' sounds like 'HARE'.
'that's' acts as a link.
(Other definitions for hare that I've seen before include "Rabbit-like animal" , "Fast-running mammal" , "As mad as this March animal" , "Leveret" , "Mammal like a large rabbit" .)