I have had an almost daily migraine on the left side since the early 80s, and though it is not a classic migraine it is extremely vicious. It feels as though nails are being pounded through every inch of my left skull and temple, and as though there's a fireball behind my eye. The migraine also stiffens my facial muscles and causes one eye to tear. I also have "cluster" attacks as though a huge sword has suddenly bisected my head. At one point, years ago, I dug sharp objects into my skull to counteract the pain; it hurt, but it briefly defeated the migraine. (Unfortunately, it also drew blood, so I stopped this. Obviously, the pain had made me very irrational.) So I have never believed that the classic migraine is the mothership of head pain.
I have heard that dental nerves can get mixed up in head pain. It was must be awful.
These days, I can't take ibuprofen due to ulcers, but back in the day, I found it quite helpful, as well as the hideously expensive imitrex. (Have you tried it?) Tiger balm, rubbed on a temple, can help, but beware -- it's painful elsewhere on the face. I also used to have success with ice packs and heat, applied one after the other.
It's interesting that your headaches start just before going to sleep. Mine start about a minute after waking.