I agree with Sharynf , contact cement is the proper adhesive. HOWEVER....
If you don't want a repeat of the "falling off" incident I suggest you take the time to properly prepare the surfaces. As you will be re-installing on a previously used surface, you will want to clean off as much residue from the previous installation as possible. While dry use a
putty knife or light sand paper to remove any residual glue or surface roughness. Once you've removed as much 'big stuff' as possible then wipe down the surfaces with Lacquer thinner. < Please read and observe all the
safety precautions that go along with the use of these HIGHLY dangerous products.> Ideally you would be replacing one piece or the other <
Laminate or substrate> altogether, but since you're probably not, you'll want to go with a thicker application of Contact cement and or
purchase a Thick Bodied or Rough Surface cement. This will help
cover the inconsistencies of the surfaces, and make things much better.
Obviously, follow all the directions on the can of cement, Wear Gloves, wear a respirator, Wear trashy shoes and a smock,
safety glasses, and do this in a well ventilated area away from any windows or other points of 'air intake'. It's not a hard project, and working with laminates can be a lot of fun but it does have risks.