I use HA at two different locations, each with their own RPI4 and config. On my iPhone, I have set up 2 servers.
When starting the app, most of the time is uses the first server, while the second is the one I use most. I can’t change the order of the sever.
I’m using Android, so maybe the functionality is different, but it always opens up to the instance I last used. Also, a simple side-swipe with 3 fingers quickly swaps between them.