I would sew a reinforcement onto the edges, sort of like what you find in the shoulder seams of a t-shirt. Measure how long the edge is suppose to be. Cut the reinforcement and pin both ends to the ends of the quilt, then the middle and then the middle of that until you have pins set only a few inches apart. It's just like putting on a border. In fact, a border of straight grain fabric may be the answer to your problem. Measure carefully in the center of the quilt and pin bountifully.