Do not cover it with cardboard or buy new carpet.
Sew it back.
What you need.
Weave thread, yes Weave thread (my "brothers" here will know what that is well) and a weave needle.
What is weave thread? It is what women, usually "the sistas" use to sew in fake hair. It is so damn thick and strong. And the needle is usually curved so it makes getting into tight spaces easy.
You can get spare material to cut out from "under the rear seats" There is usually about 2" of unused carpet tucked in under there. Thats where I took mine from.
All you do is cut out a piece to repair the spot with the hole sew it in from under the carpet. If you are skilled at sewing and can do real tight stiches, it will not be noticable. I will take pics of my "spot" sewing later this afternoon.
I have been meaning to post this for a while now, but i have been real busy (see sig)