RE: I came back to Hungary because of an injury (2023.06.07)

I didn't know you could get insurance in other places outside of where you lived

This is what the European Health Insurance card is good for. It is good in other countries. I had insurance in the past too, but that was valid only in my country (in Hungary). I have not had the European Health Insurance card, when I was on Tenerife. Now I have. It is good almost in the entire European continent. And of course I still have the local (Hungarian) health insurance too.

can you get insurance as a tourist too?

Yes. The European Health Insurance card is good for the tourists too. It is good for anyone, who is temporarily staying abroad in most of the European countries.