Hi, what is your medication. While BS in the 60s is not yet dangerous, you certainly don't want it any lower. If you are on basal insulin, and I'm assuming your dr has instructed you on how to adjust doses, then you may slightly reduce your basal insulin overnight.
What is your blood sugar when you wake up in the morning, also?
Eat protein before bed to help sustain BS higher longer. Also consider environmental factors: if you sleep with window open on a cool night, or its cold in the room, your body is working harder to keep you warm which plays a role in your BS going down. As for 60...that is dangerous low as you become immobile at 32. My experience? My friend I lived with for 5 years is a Type 1 and I saved her life... I honestly lost count, catching her at 60, 45, 32, 28, unreadable on meter when she was convulsing and slipping into a coma.