On my HH style bows, I use one piece leather. On my Adcock, I use sealskin, but I don't think I'd buy it again.

On the recurves, I use leather on theplate and carpet on the shelf. I also use elevated rests.

Are you sure you just don't need to reattach the leather? It may just need a little Barge Cement. I bet they just used leather with an adhesive side.