I don't think you can comfortably wear a coat over a poncho. You may have to just carry it with you & put it on when you get where you're going after removing your coat.
I'm wearing the marvelush poncho that I just got. Though it's cold & snowing today, I didn't wear a coat over it because I go from an underground parking lot at home to a covered parking spot at work, & it's just a short drive. True, I'd have a problem if the car broke down & I had to walk outside for a while - but that's the risk I take.