With 40" square you definitely could use a piece of house insulation foam board. Just make sure the board is cut so that the triangles on the back hold it tightly against the board. You could even use tiny, nearly invisble pins thru the front of the quilt into the foam board to hold some areas taut.

To overcome Ghostrider's concern over the weight of hanging the board with the quilt on it just be using hooks in the back of the board, how about trying this?

Jan in VA

This is a challenge. If hung on point, will any of the fabric be straight grain, up & down? If not, I think it might sag over time. So I think the foam insulation board is a good idea. It comes in 4'x8' sheets, so it could be cut to 40" sq. then the quilt could be tacked frequently to avoid sagging.
I saw a round quilt with a flange on the back. A piece of PVC pipe was connected into a circle that fit into the flange.

The Magnetic Invisible hanger allows you to hang a quilt on point https://video.search.yahoo.com/yhs/s...e0&action=view

I know how to hang a round quilt. I used a hula hoop cut open and run around the quilt. You could buy plastic tubing and run it along each edge of the quilt through a narrow sleeve sewn next the binding.