What kind of Backing for carpet ??

The carpet around the wheel wells, and footbed areas has a few loose spots. I plan to remove and reglue after a good cleaning. Should I add a bit of a stiffer backing to the carpet before I glue it back? Perhaps a heat resistant/sound proofing type?

I am not so concerned about sound, but have seen some carpeting glued back in the wheel well areas, and it loses that nice rounded look, and becomes form fitting to the shape of the wheel wells (indented a bit). Was wondering if some kind of a backing would help.

John