Just got a C6 and the adhesive on the headliner keeps giving way and it droops on my head. What's the best fix for this?
Remove the top and bring it in to an auto upholstery shop and see what they can do or suggest to fix it.
I would suspect that it has to removed carefully, and reglued using an adhesive like 3M 77 or similar.
I glued carpet material to my rear partition using 3M, and it's solid.
Not sure about the Vette but on most cars in the last 30 or so years when the headliner droops it's because the foam backing has deteriorated and separated from the fabric. Once that happens it's unlikely it can be glued back smoothly. Given that it will need to be taken down, the labor is pretty much the same as just replacing it with new material.
Probably best to take it in to an upholstery shop but if you ever want to do any of your own work. This is the best product I have ever used. It can be brushed. Once it's on, it's not coming off.
Don't overthink it. My 2007 headliner starting coming off on the drivers side about 6 years ago. I bought the 3M headliner glue in the spray can at the local auto parts store, pulled down on the headliner until it stopped coming off, sprayed it all over and pushed it back onto the top. It's been fine since then.