If the headlinings are normally glued in place as Richard says, I wonder if it is possible to inject glue through the lining at selected spots using a syringe with a wide bore needle. It would need a non staining low viscosity glue and one would have to find the right points at which to inject, but if it works it would save the effort of removing the lining.
I have no idea if it would work as I have only just dreamed up the idea. Any opinions Richard?