For example, if the disk used to be called "Data" and is now called "Backups", then symlinks I may have created would still point to "/Volumes/Data/..." but need now point to the new name instead.
Since I knew that there would only be a handful of such symlink files on my other disks, I could easily update them by hand (using Terminal.app, with the "ln -s" command).
All I needed to do was to find all those symlinks first, making sure I would not miss any.
With Find Any File, this is quite easy. Set up a search like this:
Similarly, you can also find related Finder Aliases, by searching like this:
However, if you should ever copy all your content to a new (larger) disk, file by file, Finder Aliases won't work any more if the targeted files have also been moved or their disk has been renamed.
So, it can't hurt to update your Aliases right away after moving or renaming the target item. To update your aliases, simply locate or reveal them in Finder, then select the Alias file and hit cmd+R to have it reveal its target. Should the target have been moved or renamed in the mean time, macOS will automatically update all the redundant information.