I recommend a slight variation of the Southeast Face Route, at least on the way up. It saves you from having to schlep up the steep loose scree of the Southeast Face. As you face the SE face notice the ridge immediately to its left, which is more appropriately called the South Ridge. The footing is much more secure here and makes the climb up much easier. You can return this way, or elect to plunge step down the chute on the Southeast Face on your return. I added a couple of photos to show the route.
Follow the route for the Southeast Face. However, once you are directly below the face and ready to leave the Eagle Creek Basin, head just a little more west to the visible ridge which you see directly left of the SE Face. This south ridge has a much gentler slope yet doesn't really take you that far off the more direct bee-line approach of the SE Face. The footing is much more solid. There is some intermittent sporadic low brush but this is easily crossed. Or at least it is much more easily crossed than climbing the steep loose scree on the face itself.
If you stay on or near the ridgeline itself you will eventually come to the top of the chute of the SE Face on your right. On your return you may elect to take this chute down. It makes for great plunge stepping on the downhill return. I added a photo of the top of the chute.
None, although I enjoy trekking poles while plunge stepping.
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"Working rule number twenty-three: Go through the motions anyway; you might get lucky."