Band leader in haste to become a big name (9)
I believe the answer is:
celebrity
'a big name' is the definition.
(thesaurus)
'band leader in haste' is the wordplay.
'leader' says to take the initial letters.
'in' means one lot of letters goes inside another.
'haste' becomes 'celerity' (I've seen this before**).
The first letter of 'band' is 'b'.
'b' going within 'celerity' is 'CELEBRITY'.
'to become' is the link.
(Other definitions for celebrity that I've seen before include "Fame" , "name" , "target of paparazzo?" , "Famous person seen in rebel city" , "Star" .)