We can't advise on specific dosing here so this is something you should probably talk to your team about, but we can make some suggestions and comments.
I was on lantus before going on a pump and it is a long acting insulin, so I wouldn't normally expect it to make your blood sugar drop suddenly. I would expect its value to be adjusted so that you wake up with a similar blood sugar to the one you go to bed with. What bolus (short acting) insulin are you taking?
Or are you new to T1 (LADA?) and still only taking a basal insulin?