I have some 18" recone kits ordered from Fi Car Audio and the owner of the company suggested 20 minute epoxy from my local home improvement store. He said they use the 5 minute stuff to re-cone subs at SPL competitions.
I don't know which kind of glue would be best. Might be best to call tech support at the manufacturer. If you choose wrong, you may pick something that breaks down the rubber compound of the foot. I'm sure you will want to remove the old glue before applying the new. Good luck.
The correct term is "surround". Parts Express used to sell the adhesive for this as a stand alone product. I can only find it in refoam kits like this. There are dozens of different types of foam surrounds so guessing how yours will react to epoxy would be futile. You can buy some and try it with the understanding it may eat the foam. Or you can call PE and order some of their glue for this type of operation. Your foam my be deteriorating anyway so a refoam may not be out of the question. Besides PE, there are a few Ebay sellers that have kits at half of PE's price.
For the foam to paper bond, you want to use a latex or starch based adhesive. DO not use an epoxy or acrylic and certainly not a rubber based adhesive- these are likely to damage the foam surround.
If you can find a latex adhesive, use that- most importantly, let it cure in a well ventilated area for at least 24 to 36 hours before using the speaker.