Nooooooooo... don't change how the surround mounts on the cone! Please!
The surround needs to be on the cone that way... otherwise, if you front-mount a surround that needs to be back-mounted to the cone, or vice-versa, you can preload the suspension to where the coil sits at rest, too far outward or too far inward. This reduces Xmax and increases even-order distortion (since the woofer travel will be less symmetric)...
Gary, I generally just use plain old Elmers white glue, believe it or not. On a Mach 1 woofer with its paper cone, it bonds EXTREMELY well. And if you just scrape off the loose debris from the basket, and leave a tiny thin film of the original glue on there (just sand or scrape it enough to get rid of the flakes and bumps), then the white glue will bond great to the frame as well. I've refoamed several hundred drivers this way, and they've all worked great.