Accelerate out of Athens (6)
I believe the answer is:
hasten
'accelerate' is the definition.
(I've seen this in another clue)
'of athens' is the wordplay.
'of' indicates an anagram (the answer comes from or 'of' the letters).
'athens' anagrammed gives 'HASTEN'.
'out' acts as a link.
(Other definitions for hasten that I've seen before include "Step up" , "Hurry along" , "Travel with expedition" , "Make faster" , "Be quick to do something" .)