From the saddle between the Middle Teton and the South Teton, climb directly up the ridge of the South Teton. Then turn left and enter the couloir. You can see the couloir from the saddle. The couloir ends on the summit ridge a short distance west of the summit. Generally, a stretch of moderately steep snow must be crossed in order to reach the couloir. But in 8/94, with some 4th class moves, I was able to keep my feet on rock the whole way.
From Lupine Meadows parking lot head up trail to Garnet Canyon. Cross bridge at the end of the trail and find your way through the boulder field. Once you pop out into the open, cross the small stream again and continue on worn intermittant path along the east side of the South Fork of the "Meadows of Garnet Canyon," east and right below the Middle Teton. Continue up to the saddle between the South Teton and the Middle Teton. I actually headed up the middle of the morainal debris and talus and angled over towards the Middle Teton side of the South Fork. I hadn't made up my mind if I was going to solo the South Teton or the Southwest Couloir on the Middle Teton. From Lupine meadows, car to car, it took 9.5 hrs., which included spending 1 hr. on the summit to take in the views. Great workout on a nice day.
Might want to carry an ice axe for crossing snow.