In our proposed sequence, you will return from Lakes Peak north back to Thirsty Peak before taking off on the connecting ridge to Eagle Peak. One-way mileage and elevation gain are therefore measured from the summit of Lakes peak to Eagle Peak by way of Thirsty peak. Round-trip mileage and elevation gain assume completion of the sequence.
From Lakes Peak, return NNW on the ridge that connects back to Thirsty Peak. The Lakes Peak route will provide information about this stretch of ridge, which never exceeds Class 2. As you approach Thirsty Peak, you may want to contour across the east face of the summit block about 200 feet below the summit to reach the connecting ridge to Eagle Peak and avoid going back over Thirsty. The most difficult aspect of this traverse will be the ridge that drops down to the Thirsty-Eagle saddle and climbing back up to the 12,907 ft. point. This section of ridge is very rocky, broken up and presents some minor route-finding problems. It is intimidating enough to make you look for an alternative, but there really is none. For a while, the ridge may drive you off the crest. Most of the time, we found ourselves on the north side and a little down. Typically, after allowing ourselves to be driven down, we later find that just a little extra effort to stay on the crest may have proven better. So, pick your own path here. As you approach the 12,907 ft. point, you may again be tempted to avoid going over it to save some unnecessary elevation loss, by contouring on the north side of the ridge. We found we still almost ended up on the summit of the little point.
From the 12,907 ft. point, the remainder of the ridge is a high altitude, long walk over mostly tundra until you near the summit where you'll encounter larger, flatter rocks. The views on either side of the ridge make the stroll very entertaining in regards to scenery. When we did this, we flushed out a nice covey of Ptarmigan along the way.
After you've taken a break and enjoyed the summit view, it is possible to follow the NNE ridge of the peak all the way down to Rainbow Lake. There is a climber's trail you can pick up once you enter tree, but this assumes you have left your vehicle in the vicinity of the lake. If you've left it farther up in Cloverdale Basin, then you will likely need to return along the ridge back in the direction of Thirsty Peak, though there is a report on LoJ of someone descending more directly from the summit back into Cloverdale. We returned along the ridge all the way back to the saddle or low point, working back through the rubble below Point 12,907, and then dropped north off the ridge into the upper basin below. From the vicinity of the low point of the saddle, we followed a shallow gully down that joined another gully about 1/3 of the way down. In between the two gullies was something of an "island," composed of a lighter-colored rock that was prominent from a distance and looked like dirty snow. Descend down this island of lighter-colored rock until it drops you onto an open, scree slope. This becomes the kind of scree where a few square yards may start sliding with you, but mercifully, it will all be over in about 15 minutes because after about 600 feet of elevation loss, you'll reach the basin and the nice tundra, grass and wildflowers. Empty the rocks out of your boots, then resume the hike down through the basin and try to follow the route you took up into this basin in the morning to make your way back to your vehicle and the trailhead. The best part of this return will be the wildflowers in the basin. In early August, this basin was filled with a vibrant and abundant display of aster-type flowers.
Alternate Circuit hike: Drive as far as Rainbow Lake. Ascend the NNE ridge of Eagle Peak. Follow the ridge from the summit of Eagle to Thirsty Peak, turn south to tag Lakes Peak then return to Thirsty Peak, walk NW to UN 13,123, continue NW to Cottonwood, then drop down to Silver Lake, locate the trail back to the trailhead and hike back down the road to Rainbow Lake.
Links to other information, routes & trip reports for this peak that may be helpful.