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Green Mountain is in a State Forest on the Kitsap Peninsula near Bremerton. It is the second highest mountain in the county, slightly lower than neighboring Gold Mountain. There are established trails up and around with the climax being the summit. The views of Puget Sound and Seattle are surprisingly good given the lower elevation of the mountain. The low elevation does mean you can easily hike up most of the year. There are picnic tables on top at the overlook.

Green Mountain - Kitsap County

Distance: 2ish miles
Elevation Gain: 1,100 feet
Summit Elevation: 1,639 feet
Access Rating: 1
From Bremerton follow Kitsap Way (State Route 310) for 3 miles to Kitsap Lake. Turn left on Northlake Way, proceeding for 1 mile. Turn left onto the Seabeck Highway and follow it for 3 miles. Then turn left on Holly Road, proceeding for 3 miles. Turn left (south) onto Tahuya Lake Road and in 1.25 miles bear left onto Gold Creek Road. The trailhead is in 2 miles on your left.
Access:
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The trail is multi use so you may see bikes and motorcycles. About a half mile up you cross Gold Creek and hit a trail junction. You can take the Plummer Trail left or the Gold Creek Trail right (do as a loop). At the 1.5 mile mark is the junction where the former two trails meet again. At that point, go right for a quarter mile to a trail split. There take the signed "Vista" trail up to the summit area. On top is a parking lot that is open on summer weekends if you don't want to hike this one. The Green Mountain State Forest is a working forest so access by car is never a guarantee.
Route:
Puget Sound in the distance
Summit rocks
Forested views
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DNR Trail Map Link