Going after the most accomplished wouldn't be their first preference (4,4)
I believe the answer is:
next best
'accomplished wouldn't be their first preference' is the definition.
I can't tell whether this defines the answer.
'going after the most' is the wordplay.
'going after' becomes 'next' (I can't explain this - if you can you should believe this answer much more).
'the most' becomes 'best'.
'next'+'best'='NEXT BEST'
Can you help me to learn more?
(Other definitions for next best that I've seen before include "Second in order of preference" , "Not the first in order of preference, but following closely" , "Not quite the top quality" , "Not quite the finest" .)