This is what leaders and enders are for. You don't get that wiggle if you are chain piecing, so leaders are a piece of fabric you put before the piece you want to sew, and enders are for after. Some quilting guru (Bonnie H maybe?) keeps 2" squares next to her machine, and she just grabs 2, a light and a dark, slaps them together, and uses them before or after whatever she is sewing. Cut them off, toss ina basket, and when you have a big pile of them, make them into something scrappy. 4 patch, 9 patch, whatever. You can also cut them down, if needed for whatever.
Some people keep a piece of scrap fabric and just sew over it till it's so full of thread its unusable, throw it away, and start on another.
The other thing I have found is that if I am sewing with a straight stitch throat plate (the kind with just a single hole instead of a wide slot for zig zagging) I have fewer problems with that as well as the fabric getting stuck in the slot.