To get to the South east bowl/gully you take the turn off east river road to Chico, continue on by the resort and go to the town of "Old Chico". There turn left (South East) and go about three and a half miles up Emigrant Gulch, a dirt road, to just past where the road crosses the creek. The road will start to get overgrown with willows and to climb, you can pull off and start walking where ever you feel you don't want to drive further.
The "road" switch backs up a talus slope and gets steeper and rockier then just sort of peters our. This is the one tricky part, you want to climb on up to the big bowl Southeast of the peak, but you won't really be able to tell where the peak is from this angle. Stay to your left as you go up, you should hit a place where it flattens out and a little further you are in a big bowl, filled with large scree and bounded on the top with rock cliffs. As you climb up the bowl, you'll see a narrow gully on the right (to the North). This is really the only non-technical way out of the bowl. The gully is steep and loose but it can be walked up with care. There may be a snow cornice on the very top on the gully. Climb on up and come out a short distance from the top, the rest is obvious. Great views as Emigrant peak is the most prominant peak overlooking Paradise Valley. Takes about three hours up, one and a half hours down for fairly aggeressive pace. Elevation gain from the parking area, about 4,000 feet. A great payoff for not so much effort if you're in shape.
Good boots, water. Wind gear, weather in Montana is VERY unpredictable, you could get heat, snow, rain or it might get really nasty, that said, try to keep as light as you can in order to enjoy. And stay in shape, speed is safty when bad weather rolls in.
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"Again foul weather shall not change my mind. But in the shade I will believe what in the sun I loved."
--Henry David Thoreau