Just put it in the estimate for that new feature. When your boss starts to needle you about why such a "simple" feature is taking so long, that is when you throw the bad code under the bus. You get to demonstrate a real need, rather than just a theoretical one.
The older I get, the more I believe "Big Bang" code is a bad idea, no matter what the bang actually entails. Gotta make a branch to make a feature? Gotta rewrite a large part of the underpinnings of the app to make things able to move forward? Please don't, please develop it in parallel in the mainline.
In a way, refactoring on its own violates TDD. You are supposed to write failing tests first, then write code that passes the test. Refactoring as it is described by the most zealous of TDD advocates is just a hand-waiving over writing code without testing. If we're being strict about it, write a test that proves the code needs to be changed, then you can change the code. If we're not being strict about it, what is even the point?
Regression is some of the biggest harm that can come to users, because it violates their ingrained expectations. Greenfield defects might prevent the adoption of a new feature, in which case the status quo is maintained, but regression is forward progress that is physically lost.
The thing is, many refactoring efforts make code worse, not better. Refuctoring is something I've observed many times over, by people who consider themselves quite skilled at the art. The end result of their efforts is that the code is fancier or trickier, not more robust or easier to understand.
So, from a manager's perspective, how are you to discern between a programmer whose refactoring efforts save time down the road, and one whose efforts lose time down the road. Of course, normally you have technical leadership that can make this distinction and makes sure those people get a cold hard reality check, but many organizations discourage technical people from gaining enough power that they can stop other technical people in their tracks before they mess things up.