You don't need to experience any neck pain to still have a neck issue. You can try sleeping with a pillow thick enough to hold your head and neck straight when you are on your side. If this resolves your headaches, you have found your problem.

I had a similar problem but instead of headaches, I would stop breathing during my sleep. It took months of sleeping on my back with my head and neck straight to start to strengthen my neck so I could occasionally sleep on my side. It took a year or so to get reliable stability in my neck.