There is a lot of land you do not have to pay to hunt. Gifford Pinchot and lots of timber company land just east out the Gorge. Lots of deer killed every year in the National Forest. But be prepared to hike a little.
If you have already been on a moose hunt I would Find a good guide that offers hunts on private land for elk. Elk meat is amazing and a big bull makes for a beautiful trophy.