Alright I think I cracked the code on how to give Eevee a third evolution.

>redoing one of eevee's evolution processes on it once its already evolved evolves it into its third stage, wherein it gains that type and the higher stats from each

For example, if I used a Fire Stone on my Jolteon, I'd get a Fire/Electric-type with Flareon's Attack and Special Defense, Jolteon's Special Attack and Speed, and its Defense and HP stay the same since both Jolteon and Flareon have the same for those stats.

Optimally, this would be paired with adding an evolved Eevee for every type, but we all know that we're never getting another Eevee.