If you are climbing the SE Ridge of Middle Teton, and are making good time, this variation will add 2-3 interesting pitches, in a spectacular position, to the uppermost portion of the route. (The entire SE Ridge, with or without this variation, remains a Grade III.)

As you finish pitch 7 of the standard SE Buttress route (as shown in the Ortenburger/Jackson guide, page 143) you will find yourself on a low angle section of ledges and broken rock, roughly 2 pitches below the South Summit. The "seasonal snow patch" will be to climber's right.
Walk South (climbers left) on this ledge, and you will see a prominent curving dihedral, which comprises the first pitch.

Pitch 1: Climb the curving dihedral (hand crack, 120 ft, 5.8?,) and belay in a deep chimney above the dihedral.
Pitch 2: Ascend chimney (a few loose blocks) and belay at chockstone (75 ft, 5.9ish face climbing to get around chockstone).
Pitch 3: Continue up chimney (5.9?) to a blocky ledge.
(You may be able to combine pitches 2 and 3.)

Easy climbing leads to South summit. We rappelled off South Summit in order to reach the North Summit, where you begin the usual descent via South Couloir.

I recommend this variation only if you finish pitch 7 with plenty of time and energy left. The standard Buckingham (SE Buttress) route stands on its own, and does so very well!

NOTE: the previously published description of the standard SE Buttress/ridge fails to note that the climb is usually accessed via the Ellingwood Couloir, which usually has moderately steep snow in it, and that most climbers will want footwear suitable for firm snow and an ice axe for the approach.

You can see the chimney and its chockstones from the South Fork of Garnet Canyon, if the light is correct. (afternoon sun is best) The chimney is directly under the South summit, and splits the South Face of the South summit. The first pitch dihedral is easy to spot from high on the route if one walks South on the ledge at the end of pitch 7 (This ledge is shown in the Jackson/Ortenburger guidebook, page 143)

No fixed anchors or protection.
We used one 4 inch cam; otherwise normal rack.
Remember that there is likely to be moderate snow climbing on the lower portion of this route, and possibly on the descent. Check with rangers at Jenny Lake for updated route condition information. They are usually quite helpful!

REFERENCE: A Climbers Guide to the Teton Range, 3rd Edition, Leigh N. Ortenberger and Reynold G. Jackson
Mountaineers Press, 1996, pp 142-144