I can imagine a bunch of armchair physics possibilities

normal caveats (not a physicist, chemist, lawyer, etc)

Since atoms are made of multiple components, if you can modify and measure those components individually, then it's at least theoretically possible to encode more than two states per atom. All of the following assumes you would want to keep the same atomic number for the duration, obviously if you don't care what type of atom you're storing then there would obviously be many more than two states.

If it was possible set and count how many neutrons an particular atom has (aka which isotope), then it would be possible to encode more. Even Hydrogen has three isotopes, and Xenon has nine stable isotopes (and many more unstable). Same for number of electrons (aka ions).

If there are more properties that could be manipulated for each of those individual components, then it would be possible to have even more states. (ex: electron spin).

For example, with a hydrogen atom and it's 3 isotopes, it's theoretically possible to encode 4 states (2^2, half-nibble, crumb)