I use the 3M spray on Adhesive. Its a contact adhesive, so you spray it on both surfaces, let it dry till just a little tacky and then stick em together. They make a couple different versions.
Super 77 is a good general use one and holds well. Last time I got the Hi-strength 90 to glue some mini cell foam to my dry box and a couple of Rocket boxes I use for my captains boxes and it has held up awesome through a grand trip and a Lodore trip and living on my trailer in the elements for most of the winter. I actually carry the can in my tool box now just in case I need to stick some stuff together. Most hardware stores carry both the 77 and 90 stuff.
Not sure what your sleeping pad is made of, but beware that some foams might dissolve with the solvents in most adhesives. 3m makes a version for those foams that isn't as strong but won't dissolve the foam. Its most styrene based foams where it is a problem though, so you should be fine using standard adhesives.
You should be good to go. The foam I used was cheap floor mats from Harbor Freight. A little firmer then a sleeping pad, but cheap and about the right thickness.
Either of those spray adhesives I mentioned should work great for ya. Home Depot, Ace, Lowes, etc. carry it. No tools needed...just make sure you watch where the overspray goes since its no fun to get it somewhere you don't want it.
I think I might put the soft foam on as a first layer and then pirate this old yoga mat we have laying around that nobody uses anymore and put that on as a top layer. That stuff is much denser and stretchier. And then let the sun do it's thing. I can only hope to get out enough for the UV to destroy it before I do.
I used 1" (ethafoam I believe) and applied it with elmers contact cement, using the brush that was in the can. I used an orbital sander (angle grinder is a bit more aggressive) on the lid of the box and cleaned thoroughly before spreading the glue. Spreading the glue on the foam wasn't as easy as a spray product would be, but I had no issues with overspray. Plywood and all my cast iron on top overnight. Bomber job, but we'll see how it holds up over the next decade.
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