I've read about a technique where you piece the whole top, layer the quilt, and then cut *just* the batting in three large sections. Pull away the two outside sections and the fabric from the top and back are easier to handle than having the batting in there too, quilt the middle section first, attach batting on one side and quilt that next, then attach the batting on the other side and quilt that.
This would eliminate the need to fuss with the borders...but you would eventually have the whole quilt to finangle while you quilt that last section.