I only recently realized that the outside of my cling wrap box has a white area that is tacky; the wrap is pressed against it as you cut/tear it off the roll with the serrated cutting "knife" thingy. Then that white tacky area holds the wrap in place for next time. I also only recently noticed (at the same time) that there are pre-cut tabs on the sides of the box that can be poked in to hold the roll in place inside the box. Silly that I never paid attention to those things before, both make dealing with cling wrap so much easier!
I use Glad "Cling Wrap" brand (in the yellow box) but Saran Wrap brand might have the same or similar on theirs - maybe check your box and see if it has a tacky area built in?
(And now I've posted/read/thought the word "wrap" too many times and it looks/sounds totally wrong...)