That posse goes out as quickly as possible (9)
I believe the answer is:
posthaste
'as quickly as possible' is the definition.
(I've seen this before)
'that posse goes out' is the wordplay.
'goes out' indicates anagramming the letters.
'that'+'posse'='thatposse'
'thatposse' anagrammed gives 'POSTHASTE'.
(Other definitions for posthaste that I've seen before include "Quick" , "Like Bat out of Hell" , "With particular speed" , "With great speed, like delivery of letters?" , "Hurry the mail with utmost speed" .)