Well, here's the solution.
This is the rug on the floor of my craft room/office. I simply tuck the paper under the rug. It lays perfectly flat. My rug is just large enough and heavy enough to not slip around even though the paper is under it.
If you try this idea be sure to give your rug the "slip" test. Walk over it several times to make sure the paper doesn't make it slip and slide. It probably helps that my floors are not completely smooth and shiny.