My Blood glucose starts dropping the minute my head hits the pillow, reaches a low around 4am, then rises steadily until I wake up.
I used to hypo quite a lot on Lantus, because like you, if I lowered it, I was high during the day. I solved the problem by swapping to Levemir, which I take twice a day, and I take a much larger dose in the morning than at night. I also find a lump of cheese or something with Protein and fat in it just before bed helps.
In order to stay out of the red overnight, I then have a steep rise when I wake up, which I counter by getting my breakfast bolus in quickly, and adding a couple of correction units to it. I don’t eat breakfast for anything up to an hour, if I’m having something carby, I like to see that my Libre graph has started to turn the corner and start to fall again. Or I bung in a couple of units of Novorapid, and have a carb free option like scrambled eggs.
The other option is a pump, so you can have a variable basal rate when you need it, but I haven’t gone down that route, because I have basically fixed the problem with MDI.