First check the rails without wood. If you look closely you will probably find 3 small #8 screw holes, 1 in the center and 2 more equidistant out from the center. Those holes held small round rubber circles that supported the fabric. On the bows that had the wood on the back a piece of fabric was sewn into the head liner and it was tacked in place for support.

SMS Fabrics
350 South Redwood St.
Canby, OR 97013

Makes a headliner but beware you will need additional material to fill in past the headliner down to the windows and spaces between windows and rear doors.

That is how it was on my 1946 Suburban. Let us know if yours is like this as well.
Dave Murray