Two summers back, I tried something similar: HM to Cottonwood Lakes, but I hiked to Cottonwood lake 6 the first day. The last hour was brutal, not helped by the fact I had to drive down from HM to get my permit, since they wouldn't put mine in the overnight box. So with a late start, that first day ended after sundown.
Day 2, I climbed from Lk 6 due west and hit the Langley trail. I had planned to carry my full pack up, and head west off of Langley. But I stopped short due to not enough planning and research: I wasn't sure I could get down the west-facing slope (I failed to look up the Secor book and description), and I failed to correctly read the topo maps -- you really need to haul your pack up to nearly the 4200M (13,800') contour before heading down the west slope. ...or maybe follow the contour from my topping out above Lk6, crossing over to the Major General, and then descending. Anyway, so much for plans.
Instead, I bailed from my plans, dropped down to the upper Soldier Lk from the Langley Trail due S of Major General. From there I screwed up again, (was really tired by then, too.) There IS a route from the upper lake northeast around that nose. However, I didn't climb high enough (at least to the 3460M contour), and instead saw the steep granite between the 3300 and 3400M contours. So I dropped to lower Soldier Lk, some brush crawling required. From there, I climbed out to the saddle west of lower Soldier, turned north, and hiked foreeeever to Sky Blue. Got to my campsite, tried to eat, and blew all my cookies. Altitude and exhaustion got the best of me.
Day 3 was not nearly as bad. But it was pretty long. Sky Blue over Crabtree Pass, and down to a hidden Packer Camp on Whitney Creek, about 30 min. below the JMT. Lots of small fish there, easy to catch.
From Sky Blue, I wandered to the lake west of the "WL 3697"
, but the east side of 3697 would have been more direct. Again, better research, and better use of my gps would have helped. Getting over Crabtree was relatively easy after all the previous stuff I'd hiked.
Couple of odd notes: 1. On the climb just below Sky Blue, I came upon some laundry and a water filter laying by the stream. Not a soul around, though. #2. I met a fellow wanderer west of #3697. He was looking for an "easy ridge to hike back to his camp" that a friend had described. I think he had also come up from Soldier Lk, but he couldn't tell me for sure. There is no "easy ridge" around there. Strange.
My day 4 was a circuitous route on the JMT to Wallace Creek, then up to Wallace Lk. Day 5 up to Tulainyo, over Russell-Carillion pass, and down to Whitney Portal. Days 6-7 were spent chilling in Lone Pine -- because of a fire, HM and my car were inaccessible. That's a whole different story.