He kept having to increase (3)
I believe the answer is:
eke
'increase' is the definition.
(I've seen this in another clue)
'he kept having' is the wordplay.
'having' says the answer is hidden in the clue.
'EKE' is hidden within 'he kept'.
'to' is the link.
(Other definitions for eke that I've seen before include "To -- something out is to make it go further" , "Make a laborious living" , "To -- out is to add to" , "Make a living laboriously - a palindrome" , "long ago" .)