This hike up Angle Mountain is pleasant. Angle Mountain is a relatively easy objective in the Jackson Hole area, with scenic views in all directions, including the Teton, Gros Ventre, and Southwest Absaroka ranges. It's somewhat popular, but not as busy an area as Grand Teton National Park. Many hikers here stop at a sub-summit, usually marked with several cairns. The remainder of the route follows game trails, or just passes through alpine meadows, and is a very enjoyable, and not-very steep hike up an open ridge, with incredible scenery, and a quiet summit.
Park at the Togwotee overlook (marked by a sign). The overlook parking lot is approximately 47 miles north of Jackson on the way to Togwotee pass and Dubois, Wyoming. It's approximately 9 miles west of Togwotee pass.
Hike north from the parking lot to an old fence line. Take right and head into the forest. The trail will then head north and make switch backs heading up the western side of the Angle Mountain massif. There are several trails in this area (possibly due to the nearby Togwotee Mountain lodge), but I was able to find my way the first time no problems with no backtracking. The outlined route attached to this page approximates the switchbacks in this area, using one's intuition should present no problems. The trail becomes singular higher up, and is easy to follow. At a sub-summit (probably with at least one cairn), continue east, around the south side of the next little hill, and find a game trail that heads down a couple hundred feet to an open saddle. From here continue east to Angle Mountain's summit via game trails and off trail hiking.
Standard hiking gear. Rain shell, headlamp, etc. Bear spray is probably a good idea, as the Southwest Absarokas are prime bear (both Black and Grizzly) territory.