Splitting a large class into partial classes perhaps makes life easier in the short term, but it's not really an appropriate solution to the code bloat that your class is experiencing.
From my experience, the only benefit that splitting an existing large class up gives you is that it's easier to avoid having to constantly merge code when working with other developers on said class. However, you still have the core problem of unrelated functionality being packaged into one class.
It's better to treat the breaking down to partial classes as the the very first step in a full refactoring. If you're able to easily extract related methods and members into their own partial classes (without breaking things) then you can use this as the basis for creating entirely standalone classes and rethinking the relationship between them.
Edit: I should clarify that this advice is given under the assumption that your legacy code has unrelated functionality in one class as a result of years of "just add one more method here". There are genuine reasons for having functionality spread across partial classes, for example, I've worked on code before that has a very large interface in one file, but then has all the methods grouped into partial classes based on areas of product functionality - which I think is fine.