Philosophy requires logic. It is very difficult to even conceive of how an idea could be expressed independently of logic. Most philosophy is written in language that uses logic implicitly, rather than explicitly/formally. The absence of explicit formulas does not entail absence of logic.
Nietzsche's work is, at times, an extreme example of expressing logical arguments implicitly. However, this does not entail that his arguments were without logic. As @Joseph Wiessman commented, I think that you will have difficulty substantiating your claim that Nietzsche philosophized without or against logic, without concrete evidence from his work.
Nietzsche did denounce formal logic, but: (1) this is different that denouncing logic itself; and, (2) his argument against formal logic was itself dependent on logic. You can read more about that in Stephen D. Hales's paper, Nietzsche on Logic.
I think what Nietzsche was getting at is that there is a point at which excessive focus on the granular details of logical deduction is no longer beneficial. You can often arrive at an answer that is good enough, and you don't have to have a formal logical proof to know if an argument is right or wrong. You can, but since a man's time is finite, it is best spent moving forward with the knowledge you have gained rather than validating for the sake of validation. Further, Nietzsche believed in a constant flux, which entails that what may be true at one time is not always true in the future. I consider his views on the subject analogous to Einstein's views on relativity, although Nietzsche's conception was more abstract and conceptual and Einstein's work was mathematical. Since we know from Gödel that Boolean logic can be expressed as an algebraic system and that no axiomatic system can be both consistent and complete, it seems like Nietzsche's views on the limitations of formal logic were very much in line with knowledge we have learned since his death.