It was a lifetime ago, but the '67 had arched steel rods that the headliner attached to.
More currently, my 94 'Burb has started to sag over the passenger side.
The thin cloth is very lightweight and isn't as effected by the vibrations and gravity as badly as vinyl IMO.
I'm looking for good glue; 3M makes some excellent spray ons for this purpose. I may go with vinyl, but it doesn't handle the curves and bumps in the shape of the celing without bunching up. If you have a flat interior roof without rear a/c and heat, maybe you can do it with only the corners to trim the overlap/bunched up vinyl.