I recently encountered the same problem. The paper/thin cardboard under my carpet in the back carpet area was quite warped and crinkley and falling apart in areas. I replaced it with automotive carpet jute backing, which you can buy by the foot through various online sources (I think I used one I found through EBay). There are different thicknesses available. I think I used 1/2" or 5/8". Easy to cut with scissors (I used the paper/cardboard remnants as templates) and position in place with contact cement. Does a nice job of giving the carpet a smooth but soft look and feel. Probably helps sound dampen a little more than the original paper/cardboard.
