Even if reason is the only way to find truth, that doesn't mean truth-finding is the only application of reasoning.

I'm not sure I understand. Do you have an example?

Reasoning from premises that you have reason to suspect are false, in order to see what the conclusions of those premises are? Though I suppose you could claim that doing so is trying to find the truth of the conclusions of those premises...

Mathematics strikes me as a curious case in that regard, as it's about picking axioms and exploring their consequences.

Mathematics isn't concerned with deciding the 'true' axioms, instead they explore what happens if you (say) grant/deny the axiom of choice.

You can also take the position that there is no such thing as "truth", it's just a construct created by the environment at the time. That's post-modernism for you.