J.A.
Rather than fold, roll the diaper toward the center on both sides and pin it tightly before putting on the cover. This will help create a channel for the soft newborn poo. It actually isn't unusual for newborns to soil the cover. You may just have to have a few more on hand. I would hand was them in the sink and let them dry so you can reuse them more quickly. Don't worry it gets better when "things" get more solid.