You should be able to get thin superglue from more local eBay sellers rather than StewMac. I've got several bottles of it. Beware, it has very little surface tension (so penetrates cracks easily) and so it fills nice and flat (unlike more viscous glues where the surface tension pulls the glue down into dips), but it is very, very runny. Did I say it's runny? Well it is. Very. It will run and run, so definitely tape up the rest of the neck and rest the neck off any surface so that it doesn't get stuck to it. Lots of old newspaper down to soak up the runs.

I've found it can actually take a lot longer to set than 20-30 seconds, so just leave it for a long time and don't be tempted to touch it with your fingers to check - or you may get stuck!. CA can be set off with moisture, so gently breathing on it will help. One of your humid days will probably be ideal for getting it set fairly quickly.