I haven't had any problems with folding or puckering since I started using Sharon Schambers' board basting technique. That fabric doesn't go anywhere! I also flatten with both hands as necessary while quilting.
I love that method because it works and I can do it on the tabletop in a comfortable chair.
I'm getting too old to crawl around on the floor with a bad back and wonky hips.
Here is a link to the tutorial:
https://m.youtube.com/watch?v=bhwNylePFAA