You'll have to take off your hat going through security. Jews who travel with their traditional hats seem to have cases for them, that seems like a decent idea. In a pinch, there would be no problem using a plastic bag for it, except that you'd have to wear it for takeoff and landing, or put it under the seat in front of you, which seems impractical. (Definitely don't want to get it squished in the overhead containers!)
Having it on your lap may or may not be allowed because loose items are not supposed to be left in the cabin for takeoff and landing. No problem for the rest of the flight, though. It's also best to remove it for immigration so that they can recognize you from your passport.