Most of the good areas south of Spokane are private, so acess is an issue unless you have an "in". Mt Spokane has some elk,but they are where you find them. Ive seen them North of Newman lake, out in 3rocks, really just can be anyplace. Mica is another good spot to try, but you will need the IEP permit for there and Mt Spokane. This will give you a place to start. Good luck.
Sage