When I night test I test around 2/2.30am usually, but more often if I think there's a problem.
If I got a high, I would decide whether to correct based on just how high it was. I then aim for a higher final BS than I would do during the day just to be safe.
In general, to get your Hba1c down, takes a lot of work day in day out. Test lots and correct when necessary. Pre and during pregnancy I corrected more often because my target range was narrower. I'm on a pump so can do tiny corrections.