Australian swimmer's first to get tailpiece (4)
I believe the answer is:
coda
'tailpiece' is the definition.
(thesaurus)
'australian swimmer's first' is the wordplay.
'australian' becomes 'a'.
'swimmer' becomes 'cod' (cod is a kind of aquatic vertebrate**).
'first' says to put letters next to each other.
'a' after 'cod' is 'CODA'.
'to get' acts as a link.
(Other definitions for coda that I've seen before include "End piece written for a musical work" , "Closing passage of a piece of music" , "Final movement" , "End piece written for a piece of music" , "final piece of work" .)