I'm running into this right now as a section of my vinyl headliner has fallen. You aren't going to be able to successfully re-glue it, because it's not the glue that's failed, it's the foam liner on the vinyl. The transverse vinyl strip piece seems to serve only to cover the edges of the vinyl sections. I haven't gotten into mine yet to see how these actually attach, but my guess is that they snap in place after the vinyl is applied. Basically, the old foam/glue will have to be removed from the subsurface, then new spray adhesive would be applied, followed by new foam-backed vinyl sheeting. With any luck, you can find some to match what you have in place. If I were you, I'd inspect all areas of the ceiling carefully now, and if there are any places that appear to be bubbling a little, I'd re-do those areas now. I figure it's better to go that route than to fix one area only to have another fail in another month or two and have the whole process to start over again.