Simply have to level it up to 32 let it learn Double hit, and then lvl it up and it will evolve. You should easily be able to get it to that level before the E4.
The 6th gym is around that level so you should be able to evolve it before the 7th gym.
You dont need the National Dex to evolve it, and it will register in the dex after you get the National Dex. I know that from experience.