A fun hike with little to no climbing experience needed.
This is the most difficult part of the climb. One way to approach the face is to follow the National Forest Trail 427 about 1/4 mile past Grotto Falls and then head West off-trail, cross the Creek, and find a through the dense forest and up into the basin at the foot of the mountain. This may be difficult, as entering the basin entails gaining almost 1/4 mile in elevation in less than 500 feet.
Another way to approach is from the Squaw Creek Trailhead (accessible from Gallatin Canyon). This approach involves following the Squaw Creek Trail until it turns abruptly to the south. At this point, you will need to gain the ridge to your north and follow it West for anywhere between 1/2 to 1 mile, depending upon where you ascend. Scree covers the entirety of this stage of the approach.
At this point, the basin described above lies to the east and you can either follow the ridge, summit Mount Bole, and thereby reach the base of Elephant, or pick your way down any number of chutes into the basin. The latter is very difficult considering the scree, but it allows for the establishment of a comfortable camp within the basin, whereas summiting Bole and continuing on to Elephant is both long and strenuous, as well as very exposed.
Most of the route is class 3 scrambling up a wide apron of a face. At approx. 9,600 ft the apron bottlenecks and you find yourself in a couloir with some class 4 and lower class 5 moves, but these are soon followed by more scrambling. At just under 10,000 the slope become far more gradual, and the summit is reached by crossing an arete.
It is wise to make this a two-day excursion, so camping gear is recommended. While it does make the ascent more difficult, the summit is still very attainable. No ropes or protection are necessary.