I was under the impression that there is Amtrak in White Plains, but I guess not? There is a shuttle that takes you directly from the train station (MetroNorth) to the airport, and it runs daily. I'm not sure the price, but I know it's cheap.
The only reason why I am flying back to HPN is because it's more convenient to have the wife pick me up afterwards. I fly out of LGA because there's more options when it comes to flight times - and it's cheaper as well. If I were to leave out of HPN, I'd either have the 7am flight or the 4pm flight.
Would Bradley be an option for ya?