Try using a wooden skewer to help guide the fabric at the end. Slow way down on your last bit of stitching too.
If you have a pressure foot plate with just a small round hole in it, try using that.
Also, shoving a scrap piece of fabric (folded in half) right up next to your piece and sew off on that, it will trick your machine into thinking you are still sewing on a longer piece. Butt it up snug to your piece, and sew right on to it. Leave it in your machine and butt your next quilt piece right up to it and start sewing, ending on another scrap piece.
Chain piecing can work too, but using scraps you don't have to worry about where it is sewing on the scrap and you can fully focus on your quilt piece. I have 6 or 8 scrap pieces that I always start and end sewing on.