I am not aware of a single analgesic medication that does not have the potential to cause rebound headaches. However, I can not explain your symptoms of increased pain with a single dose of narcotics. This would be unusual, and not an allergic reaction. I would recommend obtaining a headache consult with a neurologist who specializes in headaches. You may have to go to an academic center for this, and if you are in the area we have a excellent headache center at the Cleveland Clinic.
They will evaluate you and then make the appropriate recommendations. You may require a MRI of the brain as a first diagnostic test. Also, when someone is having daily headaches a medication administered daily can help prevent the headaches. The typical medications used would include some anti-seizure medications (Neurontin, Depakote, or Topamax), anti-depressant medication (Elavil, and Doxepin), and blood pressure medication (propranolol, verapamil). Good luck.