Build container "hierarchy"

[StFS]

I would like to have multiple tasks where one task creates a "base" container and other tasks create containers that extend that base container.

I'm not quite sure how to do this, is the only way to push the base container up to a registry (Docker Hub or similar) and then pull that in the other tasks? I feel I am mixing building and publishing a bit too much that way. I would like to be able to create this container hierarchy entirely locally on a developers machine without having to publish anything.