I've actually gone into the shortcomings of the two-axis political compass twice, listing four different positions that people can have that, despite common associations, can be held by anyone, regardless of where they fall on the political compass. Reality denial, bigotry, and violence are all vices which anyone of any political ideology can indulge in.
Personally, I think the third axis should be "status quo vs. paradigm shift," as in "how much do you prefer to adapt yourself to working within the system" vs. "how much do you prefer to change the system itself," or to put it more concisely, the axis of personal flexibility.
That could work. I want to change the system. So do progressives and white supremacists. None of us want the same kinds of change, though. And the traditionalists appeal to differing traditions.
Expanding on that idea, perhaps the centre of that axis should be "keep things exactly the way they are," while a return to tradition would be on one end (greater regression being farther toward the extreme), and moving to an entirely new paradigm would be on the other. Once again, however, we run into the exact same problem, because there are some things about "the good old days" (or "the bad old days," as I call them) that many of us yearn for, and other things that we'd rather leave in the dustbin of history. Likewise, some things that have never been tried before turn out to be great ideas, but other "progressive" ideas turn out to be terrible, and need to be reversed. There is one thing that I love telling self-described progressives: just because you're moving forward, doesn't mean you're moving in the right direction.
Is it change toward liberty or authority? Is it a tradition of independence or subservience? Honestly, I think the "left/right" axis is the least relevant compared to progression/regression and authoritarian/libertarian.