If you have a root account that allows logging in in text mode (no X no Wayland, no GUI), you would do that instead. These instructions are for that case. The home of root is /root , so it would not be affected.
Mount the new drive in an emty dir, if it isn’t already.
Make sure the other drives file system supports everything /home does.
Set the exact same permissions as /home/ in the new drives top level directory.
Add a line to fstab defining the other drive to be mounted automatically as /home .
Move the contents of /home over to the other drive.
Umount the other drive.
Enter just: mount /home
This should work without errors and if you peek inside, you should see user dirs and it should show up if you enter just: mount
No reboot necessary, you could just log out, switch to the GUI login and log in as regular user. After your next boot you will find out if you edited your fstab correctly to auto mount it. If not just log in as root in text mode again and fix it.