I always thought of it as an American Civil War expression, and largely, it is. This page explores some possible origins

http://wesclark.com/jw/elephant.html

Do you get the trait after first combat, or after many, or after much travel?

It is an odd expression for an ancient game, unless it actually does refer to seeing elephants...which would be a genuinely handy thing to have done, should one ever have to fight them again.