Pat,
Buy the big bottles of super glue and store it upside down, it works for all glues. It is the same method I use for storing tongue & tru oil. The air in the bottle will cure a layer of the glue, but everything below it will be fine. Also, squeezing all the air from the bottle before putting the cap on help keep the glue from curing.
I had a series of glass bows a few years back that had problems with the overlays when using super glue, both thin and gel types. I started making the phenolic and wood thinner and went to smooth-on and haven't had any issues since. I never had any problems using super glue with big horn sheep, antelope, or bison horns, just phenolic and wood. I used Zap-a-Gap super glue for years and never had a single issue, unfortunately it is no longer available.