You can't program it to continue running, but you can hit the recirc button and it will turn back on. It only will blow the fan at the 2 setting but it keeps the car kinda warm. It sounds like the button could be stuck or maybe you hit it accidentally after the ignition was turned off.

One thing I am not sure about is if the system works when the key is removed. I am pretty sure it works only in the ACC position but I could be wrong.