This is not in Washington but it's only a stone's throw from the border. Moscow Mtn is the highpoint of The Palouse Range, an 10-mile ridge that runs from near the Washington/Idaho border to north of the town of Troy, Id.
Much of the area is private timber land but access is open for recreation during the daytime. Parking for this hike is at the old site of the Tamarack Ski Area which is long gone. The access road is also called Tamarack and is plowed to about 3800' where snow enthusiasts can enjoy miles of roads for skiing and snowshoeing.
East Moscow Mtn is the site of an old lookout which is long gone as well but the views south are exceptional of the Palouse and the entire region including the Wallowas, The Blue Mtns and Seven Devils Range.
It's relatively short hike of 4.4 miles and 900' of gain but one can also visit the ancient cedar grove another mile up the road or the actual highpoint of Moscow Mtn a bit further than that for a longer outing.