So \"myLibrary\" references \"anotherLibrary\". Both libraries follow http://semver.org/
If I release a new version of myLibrary that forces consumers to update to a new
It really depends on if the public API of the main library changes. I tend to treat libraries as a black box. I don't need to know the details of how it's implemented. So unless the inner library is exposed somehow then the API of the outer library hasn't changed.
So, if the inner library is not exposed at all, I would bump the patch number and that's it. If the inner library is exposed then you'll have to decide whether that exposure has changed enough to warrant a major version bump (incompatible or breaking change).
Of course if the API of the outer library has changed to support the upgrade of the inner library then a major version bump is warranted.