Excedrin is the only thing besides morphine that cures my occasional but brutal headaches.

Do you happen to be a heavy coffee drinker? Caffeine withdrawal can cause immense pain, as I can confirm from first hand experience. For me, it starts ~24 hrs after the last coffee and it usually takes a very strong double espresso and a nap to get back to normal. Maybe Exedrin's Caffeine + Aspirin + Paracetamol has the same effect for you?

Those have caffeine in them right? I'm sure you know your body well, but that could be part of what's helping you as well; could be constricting blood vessels or something like that

Agreed. The one thing that invariably works for me is Excedrin. Not too surprising given it has all three of the most likely solutions in a single pill.