Fabulous bird to rise quickly? Not half (3)
I believe the answer is:
roc
'fabulous bird' is the definition.
(mythical or fabulous giant bird)
'rise quickly? not half' is the wordplay.
'rise quickly?' becomes 'rocket' (I've seen this in another clue).
'not half' suggests halving the letters.
'rocket' cut in half is 'ROC'.
'to' is the link.
(Other definitions for roc that I've seen before include "Vast bird of legend" , "Sinbad's giant bird" , "fabulous bird is seen" , "Gigantic bird of Arabian legend" , "Enormous, mythical bird of prey" .)