"Hades" wouldn't be in the original Old Testament, but it is in the New Testament. As you noted, it's basically how "Sheol" was translated into Greek. So in the New Testament, they continued using "Hades". That's my understanding, anyway.

It seems like Sheol was considered to be underground, though, since it referred to the grave.

https://en.wikipedia.org/wiki/Sheol#Judaism

Again, though, this is just my understanding. I'm not a biblical scholar and I can't read Greek or Hebrew.