Slightly more willingly (6)
I believe the answer is:
rather
I believe this is a double definition.
'slightly' is the first definition.
(thesaurus)
'more willingly' is the second definition.
('I'd rather do this')
(Other definitions for rather that I've seen before include "somewhat > sooner? > Quite" , "On the contrary or to some extent" , "Most certainly" , "A bit" , "Slightly, more so than not (6" .)