For spring rides where I have such a temperature change, I clean out my son's closet and wear his discarded long sleeve sweatshirts early in the ride. If it's nice, but just too small for him, I give it to someone at a rest stop who wants it (or more likely has a kid at home who will want it). If it's worn out and ready for the trash, I just toss it into a trash can when I am done with it. If you don't have a teenage son, just clean out your own closet.