Keeps staring, to an extent -- this might attract attention (4)
I believe the answer is:
psst
'this might attract attention' is the definition.
('psst!' is used to get someone's attention)
'keeps staring to an extent' is the wordplay.
'to an extent' says the answer is hidden in the clue.
'PSST' is hidden within 'keeps staring'.
(Other definitions for psst that I've seen before include "Soft sound to get someone's attention" , "Here's a secret" , "Whispered attention grabber" , "Sound of call to hear a secret" , "(Whispered) may I have your attention?" .)