Probably a beach towel or old blanket would do the trick.
You can find all the old carpet you want in the dumpster outside a flooring store. Just be careful with the carpet tacks that might still be in the old carpets.
The advantage of carpet over towel/blanket is that it's a lot heavier, so it will help keep moisture evenly in the soil better. But I don't see why you couldn't cover the old towel with mulch/leaves and end up with the same effect.