In the unlikely event this lack of forward capacity planning happens to me (cough, cough) I tend to just add another disk, move existing stuff from /var/lib/docker to that and then make it a sym-link to the new disk.
Of course that means shutting down Docker for a moment... no reboot required, though.
You may want to prune the original disk manually so its free space is reclaimed (fstrim).
Sounds easier than messing with a live CD to me and net storage consumption would be the same.