I really like Solite too. It sounds like your method is the finesse way to use it and keep it from doubling over on itself. I use a different method. Mine is more like a brute-force or gimmick method.
First I cut my piece of covering a couple of inches bigger than the part to be covered. Now with the backing side down I run a piece of 1/4 to 1/2 inch masking tape around all the edges of the top. Any cut edge that is going to be a seam I skip the tape. After taping, flip the piece over. I also use scotch tape to help get the backing off.
With the tape around the edge, the covering is now stiff and won't fold back on itself readily ... BUT if it does, it's real easy to pull apart. So just cover as normal and when you trim the edges, you trim off the making tape too.
Thats all there is to it.