Scenic cross country travel to a minor peak.
In the Flagstaff area, most people who hike a mountain will usually do Humphrey, Elden, or possibly O'Leary. It is too bad that Woody Mountain southwest of town is overlooked because it has some great views of the area from it's summit and it is a nice hike across some pretty scenic terrain.
Woody Mountain is one of the many 8000 foot+ hills that dot the San Francisco Volcanic Field. If it were north or west of the San Francisco Peaks it would be among a number of other similar peaks. Fortunately, it is relatively isolated southwest of Flagstaff and has enough prominence in the area to host a fire tower. There are an infinite number of ways to access Woody, but if you want paved access to the trailhead you'll have to come in from the east and 89A.
This description starts across from the Griffith's Springs Day Use Area. If you park on the west side of the road you should be alright, and it is free. There is a sign for Griffith's Spring. Once parked, climb over or under the barbed wire fence. There is no trail as this is cross country, and you won't be able to see Woody Mountain yet. If you head in a northwest direction, which is about perpendicular to the road, you will walk through a mile or so of recently thinned ponderosa pine that is pretty easy to pass through. After a while you will come to the end of the open thinned area and you will either pass through an area of dense forest before entering another thinned area, or you will be able to walk directly from one thinned area to another. These appear to be where section corners meet, and I think the densely forested land is State Trust Land, but I'm not 100% sure. Continuing on the NW trajectory you will probably start to see some views of the east summit of Woody Mountain. It looks like a large hill with a wide flat top. This is a view of the east summit from a small knoll (not grassy) about 1 1/2 miles away. Once you have old Woody in your sites, it is a fairly straight forward hike to the summit. You may climb and descend a few "false summits" before you finally start your way up to the true summit. You will have to pass under the power lines just before you reach the final 500 feet of hiking. You may cross one or two well maintained dirt roads during your approach, and you will meet the Woody Mountain access road near the summit. At the summit you can climb the fire tower year round and enter it if there is someone manning it. Before you leave the tower, try to study some more prominent topographic features you should be able to see on your hike back, or use a compass and get a bearing that is southeast of the tower. It can be helpful to have something to aid you on your hike back to your car as it can be easy to get turned around if you aren't careful.
There are a great deal of Elk and Pronghorn in the area. I have always seen something when out there. I would guess that hunters are pretty common in the fall seasons. You might want to wear a loud color in the autumn months.
Check out the Triplogs.
WARNING! Hiking and outdoor related sports can be dangerous. Be responsible and prepare for the trip. Study the area you are entering and plan accordingly. Dress for the current and unexpected weather changes. Take plenty of water. Never go alone. Make an itinerary with your plan(s), route(s), destination(s) and expected return time. Give your itinerary to trusted family and/or friends.