If you indeed on a full Y, I believe UA can overbook you onto their sold-out 9:10 pm flight tonight. (I know some elites get that perk. Not sure if full fare pax do as well).
Bottom line is that you are not on your own. UA has an obligation to get you to your destination, though it may take an extra day.
That said, per the above poster, it may just be easier to rent a car.