Should I write new unit tests for classes that I create during the refactor phase of TDD?

I have created my unit tests and my class and I got everything to work as quickly as possible and I have a green bar and many tests.

Now I want to refactor and make it good. Part of the refactoring that I want to do is create a new class that SUT will rely on because I realize that I had different levels of abstraction in the same class. Now the existing tests will cover the new class completely but I am unsure whether to move the tests to a new TestClass that tests the new class, duplicate them (although they will be slightly different) in both or don't bother because the original class is the external API and the new class is a private member variable.