Take a scrap of fabric and double it up (or use two small pieces). When you finish a seam, sew a a few stitches in to nothing, then sew in to your scrap. You can then cut your piece from the scrap without a tangle of threads to deal with. When you come back to your machine for the next seam just sew from the scrap piece, cutting it off when you go to press.
I usually keep two of these by the machine. One for the start of my sewing and one when I finish the seam I'm working on. That way you always have one handy to plug in to the end of your seam when you leave the machine to press.