If you’re using a single pod as a JupyterHub VM for multiple singleuser servers then if you scale horizontally you’re running multiple completely independent JupyterHubs. They just happen to be behind the same load-balancer but they have no shared knowledge.
This means you’ll need to implement session tracking yourself on top of JupyterHub and any other infrastructure, as well as a writing your own load balancer configuration to work out where to direct a new login to. You’ll also need to deal with synchronising user storage and accounts across multiple JupyterHub servers, and if you get it wrong you could end up with corrupt data. On top of all that your security team will need to review all the custom code you’ve written- they’d be better off spending the time to review the official Z2JH container images.
Failing that, you could look at something like
I haven’t used it, but others have had some success.