Breed your eevee and hatch a new one from an egg. The low level will make it very easy to train up, meaning each level will give you quick base happiness. You can use vitamins to EV train it early, and using those will also build its happiness even higher. (Use the vitamins before training, since they give more payoff when you have lower happiness, whereas the leveling up stays constant) Once it starts to slow down, you can go with rare candies if you have any left. Holding the soothe bell will also do the job.