It sounds like you have chronic daily migraines. The prescription for the mild anti-depressant sounds like a prescription for a preventative medication. You mentioned that pain killers and decongestants don't work, how many are you taking, because these could be rebound headaches as well.
I would go back to the neurologist and talk to him / her. You need a plan. If you're taking pain killers or decongestants regularly and they're causing rebound headaches you need to wean off them slowly to stop rebound headaches - but you will also need something for the pain.
There are medications called triptans that are very good at controlling migraine pain, don't have major side effects and aren't as bad at causing rebound headaches. They are used for treating the migraine.
Then there are preventative medications (like the mild anti-depressant) and taken daily. There are many types including anti-depressants, anti-seizure, beta-blockers, calcium channel blockers, to name a few. You need to work with your neurologist to figure out what's the best course of treatment. For example, I take a beta blocker and an anti-seizure medication. I also take an anti-depressant used for chronic pain which also works on the side effects caused by the anti-seizure medication.
Then there are prescription pain medications such as narcotics, NSAIDs that are stronger than over the counter (OTC) medications and are used in an emergency and are often more effective and are thus taken less often than OTC medications. For example I take Toradol (NSAID) and Tylenol 3s (Narcotic - use it very rarely now).
Educate yourself and if your neurologist won't work with you, find another one. Unfortunately there isn't a cure for migraines but it is possible to find a combination of lifestyle changes / choices and medications that can make life bearable.
Good luck. Let us know how it goes.