My "Unofficial Guide" states that you can leave your Disney rented stroller at the train station and, if you don't want to make a round trip, pick up a replacement when you get to your destination. Be sure to take with you your personal belongings, stroller name card, and rental receipt.
I'm not sure how this works though. Maybe a CM at the next station will give you a stroller sitting there without a name card? If that's the case, you may not get a double stroller.