Control BP. Morning headaches may be a symptom of not optimally controlled blood pressure, though patients have variable sensitivity to the bp. Night time BP measurements are difficult to reliably measure unless a 24 hour monitor is used. Verifying that the BP is too high at night or in the morning should cause you to follow up with your doctor to find strategies to optimally control your blood pressure.
Check for apnea. You might very well have obstructive sleep apnea. If you have over a 17 inch neck it's almost certain. You don't have a large neck, large tonsils and obesity can also cause osa. This is a very common condition so you need to get checked out for it. A number of cases of essential hypertension (high bp) are caused by this commmon problem. Treatment is fairly simple. See your doctor and ask about it.
High blood pressure. Have your medications adjusted possibly taking some at bedtime so the blood pressure may be better in the early mornings. Discuss with your doctor.