I am interested in examples of ways to know things by means that are not available to the scientific method. I strongly suspect such examples exist but I can't think of any.
Amazing. Even more so because you apparently don't realise what a huge admission it is. Other ways of knowing things include philosophy, history, sociology, experience, poetry.
I like the example of a river. Scientific accounts of a river may include: geology, meteorology, physical geography, biology, chemistry, physics and so on.
Other types of knowledge about the river include accounts that are: historical, anthropological, poetic, economic, experiential (I mean playing it in, or fishing, and so on, in case it's not clear), theological, narrative, sociological, political and so on.
Scientism is when people argue or imply that only scientific accounts of a subject are "real" or, that at a fundamental level, other approaches to knowledge don't matter, or can be collapsed in their essentials into scientific terms.
But it takes a pretty stupid person to argue that historical, poetic or political accounts of a river either don't matter or that they can be reduced to science.
And scientific reductionists should watch their back too, because reductionists don't tend to want to stop at just eliminating non-science from knowledge. Even within science reductionists insist on a hierarchy, saying biology isn't really a hard science, and that chemistry is really physics in its essentials.