Avoid publicity during the night before (5)
I believe the answer is:
evade
'avoid' is the definition.
(I know that evade is a more specific form of the action escape)
'publicity during the night before' is the wordplay.
'publicity' becomes 'ad' (short for advertisement).
'during' is an insertion indicator.
'the night before' becomes 'eve' (eve can mean the night before something).
'ad' placed inside 'eve' is 'EVADE'.
(Other definitions for evade that I've seen before include "Dodge (taxes)" , "Dodge, steer clear of" , "Circumvent, dodge" , "Slyly escape" , "To get away from" .)