There are a few things you can do to ensure you are not cutting off the supply of blood with your efforts. First, slip a couple of fingers between the bandage and the victim. If you can't, re-wrap it looser. If the white is promptly replaced by blood, circulation is good. If unsure, compare with an uninjured limb.
Elastic bandages are useful for injuries less common than sprains, too.
If you don't have a sling or triangle bandage the sleeve can be safety pinned to the front of the injured person's jacket. A pack's padded hip belt can also help hold the hand onto the body. Finally, wrap an ace around the upper arm and the person's body, under the opposite arm, to prevent outward movement of the elbow that would hurt the shoulder. Any large muscle that is pulled can benefit from compression. While skiing through a thin snowpack, I once snagged a downed tree branch.