Driver having to edge skyward (5)
I believe the answer is:
pilot
'driver' is the definition.
(I have seen 'Driver of aircraft' mean 'pilot' so perhaps 'driver' could also mean 'pilot')
'to edge skyward' is the wordplay.
'edge' becomes 'lip' (lip is a kind of edge).
'skyward' is a reversal indicator (in a down clue, letters go up).
'to'+'lip'='tolip'
'tolip' backwards is 'PILOT'.
'having' acts as a link.
(Other definitions for pilot that I've seen before include "Airman" , "Lead" , "experimental programme" , "Small flame that ignites a gas boiler" , "Flier of aircraft" .)