I've been reconing speakers for 20+ years and lately, since my old supplier is no longer in business, I have been using 3M #847 gasket adhesive. This stays pliable so you don't need to worry about the glue getting stiff and cracking and it gives you a bit of time to get the cone positioned correctly if you move it a bit while applying the glue. It comes in 2 colors, yellow and brown. I use the brown. Also, this is good enough to not worry about getting all of the old surround off. Yes, it needs to be as clean as possible, but a little here and there won't make a difference. I have used this to attach spiders as well.
There is special glue for attaching the cone to the spider and the dust cover to the cone though. Don't use this stuff for that.
Dan