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One of the shortest trails to an Olympic Mountains summit is the Mount Zion trail. It isn't anything special, the hike is mostly in conifer forests until near the end where views are decent. I like that Mount Zion is well north of the majority of high summits when viewed from Seattle so it is very identifiable.

Mount Zion

Distance: 1.8 miles
Elevation Gain: 1,350 feet
Summit Elevation: 4,278 feet
Mt. Zion Trail #836 starts 13 miles northwest of Quilcene on Forest Service Road 2810. Take the Lords Lake Road 1.5 milesnorth of Quilcene on U.S. Highway 101. Follow Lords Lake Road to the lake and turn left onto FS Road 28. Continue on FS Road 28 to Bon Jon Pass. Stay on right at Bon Jon Pass, drive approximately two miles to trailhead.
The Mount Zion trail ascends the southwest slope, is well maintained, gets there quickly and is easy to follow.
Washington Climbing
Mount Zion summit Mount Zion view Mount Zion Map