Cut 1" strips of a complementing fabric (or the same fabric) and sew them to the 2" binding strips. This works well, although the binding will be somewhat bulkier.
I did this with 2" scraps from my grandkids button down shirts sewn together. I then added 1" strips of the backing fabric to the scrap strips to stablize the scraps. The scrap side was turned to the top of the quilt. As I said, it was bulkier, but it looks pretty good and was perfect for the quilt.