Al, so sorry this has taken me so long to answer. I’ve been mostly under the car the past week. To answer your question depends on what you mean by the glove box. Do you mean the actual box itself or the entire assembly that forms the whole right side of the dash panel. I’ll do both.
The glove box is not supported by any attaching point underneath it or on its back. Here is the glove box (the whole panel) and where it fits in the my car, a '63 FHC 887030. As far as I can tell my glove box is original.
The attachment of the entire glove box face (the dash panel) is on its left side via a bracket that is secured on its left to the partition of the fuse area, and by two bolts on the right side fitting into a dedicated bracket fixed to the car body and, of course, this makes the entire front of the glove box to be metal. The actual glove box itself inserts into this metal facing from the rear and is attached to it at the front bottom with two screws and at the top by slipping under a metal bar. That metal bar is fixed to the front facing with 2 nuts on small threaded studs. There is no attachment of the actual box at its rear. Pictures tell the story I hope.
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Hmmm the website has stopped uploading some of the pics and logged me out. I’ll load them again
*Jim63EType if you have '63 coupe, I’d sure like to see pics of it as I’m putting mine together, but I did not take it apart. Little details have been a challenge.