Have you thought about switching to using cloth bags for your gift giving? I see after Christmas a lot of wrapping paper and materials in the recycle bags. It is fun to rip into paper-wrapped gifts, but the amount of waste is mind boggling (even if they are supposedly "recycled").
Simple string bags are easy to make, and you can ask the bags back to be used again and again. Or you can ask the gift receivers to use them for something else.