That loop is a very good way to do it. And camping the first night at Chicken Spring Lake per Ken's suggestion is a great way to start the trip. Going in on the Cottonwood Pass Trailhead, you'll need to change your permit. When you hike out at the Cottonwood Lakes Trailhead, just before you get to the trailhead, turn right on a short connector trail to the Cottonwood Pass Trailhead and your car. Be aware that the location of the Cottonwood Lakes trail is incorrect on the USGS topos (out of date) for the last couple of miles to the trailhead, and by extention on GMAP4 and ACME, so don't let that confuse you. If you happen to have the Tom Harrison Golden Trout Wilderness map, it's correct on there, but if you don't have that map don't worry, its easy to find the way.
The stretch of trail from Chicken Spring Lake to the junction with the New Army Pass Trail is dry so carry the water you'll need. Hiking that route, it's very cool to look out across the lower terrain as you go along and then watch the view of The High Sierra open up in front of you. Very dramatic.
