I like my wrapping paper on rolls, rather than folded, because it stays nice and smooth with no creases. But, I have usually kept a partially used roll of paper neat by putting a little piece of tape on the loose edge. The bad thing about that is when you go to use it again, it often rips where the tape was attached and you waste paper! I have also tried a rubber band, but that can leave its own crease.
A super easy storage idea for wrapping paper rolls that keeps your rolls of wrapping paper neat and together is to use an empty toilet tissue tube. Just use a scissors to cut through one side of the tube lengthwise. The cardboard should have enough “spring” in it that it naturally closes itself around your roll of wrapping paper.