Prune your "Silver Moon" clematis immediately after the bloom period ends and certainly by the end of July. This timing will prevent buds for the following year's bloom from being lost during pruning.
Cut back any dead, broken or diseased-looking vines, taking them back to a point of healthy, live wood. Carefully pull all of the cuttings out of the plant mass to prevent them from decaying and causing infection and insect problems.
Prune back the vines as needed to keep them within their defined growing boundaries. Remove up to one-third of the plant's volume if the plants are very overgrown for the location or need rejuvenating.