When people say the phrase "other ways of knowing", they almost always refer to spirituality or religion as an alternative to science. I don't think spirituality or religion are valid ways of knowing. That doesn't necessarily mean they have nothing to offer, it just means that among the things they offer, truth is not one of them.
But of course, that doesn't necessarily imply that science is the only way of knowing. I recently took a class that compared evidentiary practices in different disciplines, and I have been firmly convinced that science is not the only way of knowing.
Case in point, a court of law is a way of knowing whether the defendant is guilty of some crime. A court of law involves lawyers to present the evidence, juries to make inferences from the evidence, and judges to oversee the whole process. The evidence consists of testimony, writings, and objects. None of the people involved need be scientists.
Some of you may feel that the imperfections of law somehow disqualify it as a "way of knowing". Think of O.J. Simpson, who was acquitted of murder, even though we're pretty sure he did it. But who among us is so naive to say science is perfect? Personally, I feel that the scientific discipline follows higher standards than law, but it would be quite arbitrary and self-serving to pick a standard which is just low enough to qualify science as a "way of knowing", but just high enough to disqualify law.
Another possible objection is that a court of law is merely practicing a kind of science, if not the formal discipline of science. Perhaps so. But we say this in the same sense that we say toddlers are "natural scientists" whenever they tear wings off flies to watch their reactions. As much as I love framing kids as little scientists, as if all humans had a natural capacity for science, it makes about as much sense to call them "little lawyers" or "little jurors". Everyone has the natural capacity to be a juror.
So if anyone asks you if there are other ways of knowing besides science, you can say, "Yes. A court of law is another way of discerning truth."
Now that we have a specific example of "another way of knowing", we can get the ball rolling and think of more examples. How about history? And journalism? As much as we may criticize journalism, it is surely better than random guessing.
I propose that skepticism is another "way of knowing", strongly influenced by science, but different from it. Skepticism has informal rules about how to find evidence, how much weight to give different kinds of evidence, and how to make inferences from that evidence. Unlike the other disciplines, skepticism is a discipline for lay people. You don't need to be a scientist to do it, except in the sense that we're all scientists... in our hearts.