So, a series of tips.
Read up on the nitrogen cycle. Running a tank for a while doesn't automatically equate to it being prepared to handle and house livestock. If you don't have one, invest in a liquid test kid. API makes a freshwater master test kit that's ~$20 on Amazon. If you have any media you can move from the 5, or even just that filter and run it concurrently for a while, that'll be of value while the 20 builds up a colony of beneficial bacteria.
Live plants, generally speaking, or better than artificial (since I'm not sure what you have in there) because artificial plants often have hard points that can damage a betta's fins.
Cories are commonly accepted as safe tank mates, as are some snails. That said, bettas have their own personalities. Mine has largely ignored the ghost shrimp in his tank, until he decided one had to be dead. Now, I'm wondering if the 4 others that died did so without his intervention, and if not, when the last one will be killed. So far, he's ignored the snails. He chases, well, follows, his ember tetra roomies. Even when a betta is seemingly okay with a roomie, he may decide one day that he's not. One way to help with the introduction of new roomies is to rearrange the tank, so the territory is different, so he won't feel it's quite as much *his* and will be more tolerant of interlopers. Just stay away from fish that'll remind him of himself, like skirt tetras, guppies, and the like, and you'll have better likelihood of peaceful interaction.
I know you didn't ask, but if given the option, I would go for a 20 long in the future, rather than the 20 high. It gives you a better footprint to make for more horizontally complex scaping, creating more visual territories for your fish. The drawback is tall scaping doesn't work as well.
While I think your betta would do just fine in a 5, I'm glad to hear you're moving on up to the 20, so you have room to give him some potential friends.
