Lay the rug outside on a sunny day. The sun will heat the rug and help it to lay flat. Leave it outside for several hours and periodically check the progress. Lay a blanket underneath it or hang it from a clothesline by clothespins.
Place carpet tape on the underside of the rug in the position it should be in and allow the tape to help to stretch it out. Tape the edges and the middle of the rug. It helps to walk around on the rug after it has been taped to help it stick.
Steam clean the rug and hang it over a banister to air dry or allow it to sit outside in the sunshine.
Hang the rug in the bathroom and turn on the shower, allowing the steam from the hot water to steam the wrinkles out of the rug. Dry the rug by hanging it outside on a warm sunny day. Use clothespins to hang it on a clothesline or lay it on a blanket on the lawn.
Place a clean dry white towel over the wrinkled areas and iron over it on a low steam setting to remove the wrinkles.