I think the binary Hamming $(7,4,3)$ code is far and away the next best step because it has this simple visualization using three circles that you can find at the Wiki article. It doesn't require any background besides being able to do binary arithmetic. (Incidentally, my advisor was fond of calling this the "Mickey Mouse code" both because the circles can be arranged to resemble the character and because it can be taught to elementary schoolers.)
This really helps to graphically drive home how the parity bits interweave the information from the information bits. It also helps students get a better feel for how erronious bits can be corrected and detected by the redundancy. The next best thing about the binary Hamming codes in general is that they have easy-to-understand and easy-to-use parity check matrices.
If you want to take a break from binary codes and explore codes under different moduli, you might consider going with some of the simple codes with parity check matrices $[1,1,\dots,1]$ or $[1,1,\dots, 1,-1]$. They are only very simple codes with a single parity check, but they are used in a lot of places (none of which I can remember instantly.)
It also makes it easy to segueway into codes with slightly more complicated but still 1-dimensional parity checks like $[1,3,1,3,\dots]$. I'm not a big fan of For all practical purposes but it does contain a lot of nice examples along these lines from everyday life, like UPC codes and ISBN codes.