Starching the backing fabric prevents puckers. I heavily starch mine using a 1:1 solution of Sta-Flo and water. Heavy starching prevents the fabric from stretching or moving around while you quilt.
Since you have already re-layered your quilt, you could spray starch both backing and top several times. Lay a large flat sheet on the floor, spray the sandwich with starch and let dry, spray again and let dry, etc. then turn over and do the same on the other side.
Here's how I starch my backing fabrics. Mix a 1:1 solution of Sta-Flo liquid laundry starch and water, "paint" the mixture onto the fabric using a large wall-painting brush until fabric is saturated, toss in dryer, iron with steam. This is a very fast way of preparing the fabric, which ends up with the stiffness of lightweight cardstock. Don't worry about the stiffness; this is what prevents puckering while you sew.
It also helps to starch the top before layering, but for that you are limited to spray starch.