Those who worry about the security of the quarter sheet when not using the girth loops might feel better after viewing photos or videos of Thoroughbreds during morning gallops where a folded quarter sheet stays put even at speed.
I always recommend using a saddle pad under your quarter sheet. Saddle pads are much easier to clean than most quarter sheets, especially if your quarter sheet is made of wool, and your horse will always be happier in clean equipment. The area under the saddle will accumulate the most sweat and hair, needing more washing than the areas covered by the quarter sheet alone.
Don't use your thickest pad as the quarter sheet will add a small amount of extra cushioning and you don't want to alter your saddle's fit.
Start by placing the saddle pad just ahead of where the saddle will eventually sit, where you would put it if you weren't planning on using a quarter sheet,
Lay the quarter sheet over the saddle pad, lining up the front edge of the saddle pad with the front edge of the quarter sheet.
Place your saddle over the quarter sheet, smoothing any wrinkles so that they lie behind or beside the saddle rather than under it.
Lift the front of the saddle and fold the front corner of the quarter sheet back under the sweat flap, ensuring that the seam does not end up under the panels where the rider's weight could make the bulk of the seam uncomfortable for the horse. If you would like to be able to move your leg further back, simply change the angle of the fold.
Lift the sweat flap on each side to check that the fold is smooth, without wrinkles.
Check the saddle placement. If it needs to be moved back, grip the back of the saddle pad through the quarter sheet while shifting the saddle with your other hand to ensure that everything moves together without bunching.
All that's left is to attach the girth and everything will be held together by the pressure of the saddle and girth together. This set-up also allows the saddle pad's billet straps to be used if desired because the quarter sheet is folded away from that area.