N.B.
I am a migraine sufferer. My neurologist says I should almost never take regular pain relief (Tylenol/Advil products..their generics, etc...your analgesics), as they are famous for the rebound/bounce-back headaches. I was told the same as you..once stuck in that cycle, you have to quit taking them and wait it out. Miserable.
I only take my prescription meds now. I have different degrees of pain relief to take. 2 are the same...but for the ingredient of Codine. So at first or for a mild headache of ANY kind I take the milder form without the Codine...if it gets worse or simply doesn't go away, I take the jazzed up one. If still in pain, I have a muscle relaxer one to take. Its the pain med cocktail that works for me.
I also take a preventative "cocktail" of meds nightly. Migraines suck, but with this extreme regiment I have gone from 15-18 headache days a month to 1-4....much better quality of life for me.