(If you think about it, we usually leave a tail of several inches when we cut the thread, so the needle doesn't come unthreaded. This ends up at the beginning of the next seam. If the seams you are sewing are that length or shorter, the majority of your thread ends up in the garbage, it really does add up! And saves bobbin winding too!)
OR...and this is even better, instead of a scrap, use two squares of fabric and piece them with a 1/4 inch seam. The pieced parts of this quilt top have all been done this way, and since it was made with scraps except for the neutral background, it cost very little time or money! (I will say that hand appliqueing with batiks is not a lot of fun though, it's very densely woven...that's how the flowers were done)
My top still needs the rest of the borders, planning to do three, hopefully sometime...I am working on a pieced border for the second one using this trick too.