I like to cut borders LOF and I will often just slice up the rest of the fabric into 2-1/2 strips to be used later for binding, narrow borders, or cut down smaller for sashing, etc. I love my super long binding pieces!
I also save trimmings from quilt backings and cut those into long strips of 2-1/2 wide, for the same purpose. I always have all kinds of binding fabric pre-cut to choose from, much of it in very long strips. I do a lot of scrappy quilts so it's fine with me if the binding fabric is not anywhere else in the quilt.