I wake up the same way and after YEARS of this. I was told to talk to my Dr about this last night.
Day 3 of a score of 10 in the pain scale and I was in the hospital the Dr on call said it really sounds like this.
Occipital Neuralgia - I've heard of just about every headache out there but this one. I asked if this could be why even my pillow causes pain and he said very likely.
Check it out and good luck. I know how painful this is.
I have looked into the dentist about clenching teeth and that and that does not apply for me.