Starting from the same trailhead at Needleton, climb past the stream crossing, leaving the trail at about 9.800' to start your mile-and-a-half of bushwhacking towards the basin at about 11,700' immediately west of the peak. This is one of the most desolate and amazing basins I've ever camped in, ever. Windy as hell when I was there, but worth it.
Someone in your group has got to be an experienced bushwhacker to negotiate this stretch, using a topo map and compass. It contours above and to the North of the North Pigeon Creek for about 1.5 miles, ending in a huge basin directly below Pigeon's Western countenance, flat as a football field, and rarely visited. I'm intentionally not trying to detail the route, cz if you've bushwhacked over distances before, you'll know how to set your course. If not, go the standard route. This one is obscure and worth finding.
As late as the end of June, this takes you up the same North-face approach as the 'standard' route, so take an axe, and possibly crampons. May or early June would make a great snow cruise. To avoid snow, go in late July or August.
If you have information about this route that doesn't pertain to any of the other sections, please add it here.