Refactorings worthy of the name aren't scoped by class. In fact refactoring is one of the strongest arguments against unit tests; refactoring typically changes the split of responsibilities and alters the articulation points in the design, such that old tests are discarded and new kinds of tests need to be written.

Solid integration tests may work, but it's hard to get really good coverage in any reasonable running time from integration tests.

These days I try to cover the happy path with a fairly integrated flavour of test, the edge cases around the tricky bits of code, and fairly exhaustive coverage for authentication / authorization code paths, and not a whole lot more.