Tangen Tunnel is an adventurous route to the summit of Green Mountain. It ascends a "tunnel" of rock that gives the route it's name. You will experience some horrific bushwhacking through downed trees, thorny brush and poison ivy. That aside this really is a great route. Solitude abounds and a sense of adventure not easily found in such a crowded local. You will get up and close with the Fourth Flatiron, Fifth Flatiron, the Tangen Towers and Schmoe's Nose. You will also get the chance to go over East Green Mtn on route to Green Mtn proper.
This route can be followed just to climb the tunnel and you can leave the worst bushwhacking for more hardy souls.
Park at Chautauqua on Baseline. If the lot is full there is usually street parking within walking distance. Gregory Canyon parking as well as other parking locations could work but I will describe from Chautauqua since it's free. Follow the Royal Arch trail to Sentinel Pass (it is well marked). Continue down towards the arch until the trail starts uphill again. You will go up some steep wooden steps with a neat little cave/alcove on the right and then up some rocks that flow with water at certain times of year. Look sharp here. Past a slab (the start of the Fourth Flatiron) and before the next rock (Lower Tangen Tower) there is a gully that leads up. This is Tangen Spring and water comes down here part of the year.
Head up between the rocks and enter the cave. At the back there are a series of steps that spiral up and exit the cave higher up. This is the Tangen Tunnel and is a neat place to be. The "steps" are class 4 maybe 5.0 (the last one has a bit of exposure) but are very doable without a rope. The exit looks like a small hole from above.
Once above the tunnel, there is an awesome view of Royal Arch higher and on the left. From here you are between Lower and Upper Tangen Towers and can get back to the Royal Arch trail by simply hiking down the other side (south side) of Lower Tangen Tower.
To summit Green Mtn, simply find a reasonable route through the brush and work your way up. Stay between the Fourth Flatiron (on your right) and the formations on the left. The route is up to you here and have fun with that. There are two gullies that branch off right. These two separate the three pieces of the Fourth Flatiron and you should not hike up either of them. Once you have battered up the gully and have reached the ridge crest rejoice, the worst brush is below you. Now work your way uphill and north (right) to the summit of East Green Mtn. From here simply pick a line towards the actual summit due west of you.
From the summit of Green Mountain you could retrace the route but that would be miserable. A better way is to follow the Greenman trail down to the north. You will be going the opposite direction of your car but it will work out. Eventually the trail will reach a junction with the Saddle Rock trail. Take the Saddle Rock trail towards Gregory canyon and simply follow the signs back to Chautauqua (or Gregory Canyon if you parked there). You could also take the climbers access trail up towards the First Flatiron form the Saddle Rock trail and follow it down the front of the First Flatiron back to Chautauqua as well.
Usual hiking essentials.
Long pants would be a great idea and maybe some bandaids.