You might have to stop by an auto upholstery place in your area to see what they suggest. Most of these headliners were made of a cotton fabric that over time and moister would stretch and deteriorate. If it's fragile it might not take it, but the only way is to unattach it and pull it tighter. These were usually put up with nails so depending on how fragile the fabric is now whether it will stay in one piece.
Where Are You From? Oviedo, Florida
Do You own a car built by Durant? Yes 1928 Durant Model 65 4 door sedan
Thanks Mike --- My car is on an island in Maine and runs like a top. Never restored. Am very worried about how fragile the fabric is. Perhaps I can attach something to the sides to gather the fabric and stretch it gently.
Where Are You From? New York
Do You own a car built by Durant? 1929 Durant Model 666
Alex, the cloth is rotten. All you can do with it is close the doors and Do Not Disturb.
Now, if you want to enjoy and drive the car, replace the headliner and all the windlace and be done with it. Besides, it will smell better!!