"The Jaw" is the second-highest (30 feet shorter than St. John) of the series of peaks and spires that surround Hanging Canyon, between the more popular Cascade and Paintbrush Canyons. Although the area's most popular among technical climbers for routes up Symmetry Spire, Baxter Pinnacle, and other crags, The Jaw offers a moderate class-3 climb on snow or rock (depending on the season) with little to no exposure, and a gorgeous approach hike through a canyon past high-altitude lakes and cascading waterfalls. In addition, the canyon offers beautiful campsites and the opportunity to set up a camp and bag some of the other peaks in the area. The summit itself is not the most visually attractive, especially compared to its neighbors, but the scenery offered from the summit, and the relative ease of the climb make it a good introduction to Teton mountaineering.
The trail up Hanging Canyon is accessed from either the Jenny Lake Trailhead off of Jenny Lake Road or from the String Lake Trailhead off of Teton Park Road. The South Jenny Lake area is where you'll find a visitor center, campground, store, and ranger station, as well as a shuttle boat that can (for a fee) can cut some distance off your hike. Jenny Lake is about 25 miles North of Jackson, the closest city, and 10 miles North of Moose Junction (food, gas, gear shop, another visitor center, post office).
There is no maintained or signed trail up the canyon, but a "social" climber's trail does exist until about treeline. The trail can be found heading West a few hundred feet North of the West shore boat dock. The trail (and the dock) are 2 miles from South Jenny Lake parking or 1.7 miles from String Lake parking. If you have $10 to spare ($7 one-way), a 10-minute boat ride can replace an hour of walking. Once on the trail, you'll start to ascend steeply through forests and open grassy areas. The trail passes the base of Ribbon Cascade, then switchbacks up the steep first wall of the canyon. After a few false summits, the trail reaches Arrowhead Pool, and Cube Point and Symmetry Spire will be visible to the South. Make sure to bring sturdy boots, because the trail can be covered in loose rocks and dirt, and at times is more of a scramble than a walk. The trail hugs the North wall of the canyon up to Ramshead Lake, and then it's one more small hill until Lake of the Crags, which is the destination for most day-hikers.
Around Lake of the Crags, you'll probably lose the trail among talus, snow, or both. Even in late June the lake was still frozen and most of the face was covered in snow. An ice axe is necessary except possibly for a few weeks late in the season. Stick to the North (right) side of the lake, as the south side is very steep and prone to slides, and it would be a cold, wet landing if one were to slip. Be careful of moats created by melting snow, especially during the melt season. Be especially mindful of any seasonal melt streams that may be hiding under the snow. The route up the mountain is never very exposed; the main dangers are either punching through snow onto rocks, an uncontrolled fall on steeper snow that would end up in rocks. Check with the rangers about snow conditions, wet slides are a possibility, and the snow will probably be punchy after noon during most of the summer.
Either way, the entire route is visible from the West end of Lake of the Crags, so just choose what appears to be the path of least resistance. Stick to the snow if possible, avoiding any sections that look like they would hide rocks or streams. There are 2 or 3 bowls that are good opportunities to re-evaluate your route. At the final bowl, I found it easier to scramble up to the right on some rocks to get to the summit ridge, then traversed the rest of the way to the peak, rather than climbing directly up. The actual summit is the unassuming pile of rocks marking the end of Hanging Canyon. Be careful at the top, because one wrong step on a cornice or loose rock could mean a long fall down the cliffs on the backside.
Descend the way you came, continuing to watch for rocks hidden in the snow. An ice-axe glissade can save time, but only if you know what you're doing.
Camping and Red Tape
The Jaw is located within Grand Teton National Park, so you'll need either a parks pass, or to pay the $25 entrance fee. The $25 fee is good for 7 days in both Grand Teton and Yellowstone parks.
There are a few bivy sites located next to the high lakes that can be used as a high camp. A free backcountry permit is required, and can be obtained at the Jenny Lake ranger station the day before or the day of. A bear canister isn't exactly necessary, but might be handy, if only for the marmots (of which there are plenty).
For day-use, no permit is required, but it's a good idea to tell someone where you are going and when you plan to return. It's a good idea to stop by the ranger station for conditions and advisories even if you aren't getting an overnight permit.
Car camping in the area can be found at Jenny Lake (fills early), Signal Mountain, Colter Bay, Gros Ventre, Lizard Creek, and Flagg Ranch, as well as primitive sites nearby in the wilderness/forest areas. Rustic cabins and good local beta are available at the AAC climber's ranch.
Post-climb, I recommend heading to Signal Mountain Lodge for a pile of nachos.