My son bought me this magnetic door hanger at Canadian Tire. I don't know who would stock them in the US? I have a metal steel front door so it works great however it has 2 section even though I just use the one on the door. The 2 sections are made to go back to back on a front glass door too. I use the magnetic one for my Door A Doodle selection every month.
If your screen door has some metal in the top that is magnetic, look for a magnetic towel rack. They have them at Walmart for on your fridge or stove? You could put it on the top inside the door and hang the flag facing out?