I think the best way to do it is just to start an unbound container and point Pi-hole upstream to that container.
This is an example of Compose file using the default bridge network: unbound-docker/examples/pi-hole/docker-compose.yml at b8bb4f2f6b02ade931c2f0064678c368c3f6d936 · klutchell/unbound-docker · GitHub.
Just change the Pi-hole volumes to match your current configuration (this will keep your current settings when you start the new container) and adjust your environment variables (TZ, password, etc).