Over Piute Pass destination suggestions?

Depends on if you are doing miles/passing through, or fishing.
Some great X-country routes south - Goethe (Alpine Col?), Paine, Packsaddle, Honeymoon but...
prefer going north - Mesa, Desolation, Puppet, and then cross French Canyon and up to the Royces.
A great 10+ day trip...non-loop?  - up Piute Pass and down Pine Creek with an overnight stop in Granite Park.

If fishing for Goldens...do a X-country loop around the Bears...Piute Creek to Turret, and out via Italy Pass.
Amazing country - stark/no trees - seldom visited.

No doubt the usual mindset of advice for a week in this basin will be a lake-to-lake-to-lake-athon with at most a single layover. My suggestion as a photographer is to consider the best background peaks at the different times of day and site a primary base camp plus one or two one-nighters to spots best camped at instead of day-hiked to in order to best capture those features.   Humphreys is not a place one will enjoy if weather is windy as is common or if  the Mexican Monsoon is visiting because it is all so exposed.  If so ignore the below and hide down in the forested canyon.   Or more wisely don't even bother if forecasts appear to be so and go somewhere else as Plan B.  

The area between Mesa, Square, and Tomahawk is a good central zone to set up a 2 or 3 day base camp in order to quickly reach several reflecting lakes and small ponds. During mornings there are good reflections towards Glacier Divide to the south that one does not have further east in Humphreys Basin. And those same waters late afternoon can include the Sierra Crest areas including Humphreys however waters are likely to be wavy. And in late afternoon the Humpheys Lakes are only an hour's walk so one is close enough to walk back easily at dusk. Camping about those upper lakes is rather rocky, sterile, windy, without anywhere to hide from the searing sun while down at the lower areas has areas of whitebark pines one can ID on Google Earth.    Frankly, there are few comparable areas in Sierra high country that one can reach with day hikes of less than a couple hours that offer so much.  A fine place to get to know at a relaxed pace instead of continually traveling about lugging a pack and dealing with setting up and tearing down camps.

Additionally from that base camp mid to later morning when looking down on lakes is best because the water will be more blue, one can easily climb up to the French/Piute divide for views of the French Creek bench lakes with craggy peaks north in the background. But in any case there are some smaller basins within the zone that one will also want to camp at for single days. My choice on Glacier Divide is the Packsaddle Basin. To the north I like the largest of the Royce Lakes. And for your last night before escaping back over Piute Pass if skies are clear enough for alpenglow, one will find the lakes just southwest of Humphreys best as the big peak receives the latest light and its rugged character will remind one of the Dusy Basin crest.

David
www.davidsenesac.com/2015_Trip_Chronicles/2015_Trip-Chronicles-0.html

I'd say spend the first night in Humphreys Basin and see what happens after that. There's so much I want to explore in that area to the north and south.

I've never been over that way to Granite Park etc. though everything I've heard about it echoes your opinion. Need to add that to the list.

Sigh. Damn job.

Yeah at those elevations the sun matters. I was puttering around in a t-shirt and shorts one morning in the sun around Guitar Lake west of Whitney and glanced at my little gimmick thermometer I had to see the overnight lows. Well I was amazed: 14 degrees! I was just a tad cool but nothing uncomfortable, decided the bright July sun coupled with the dead still high dry air like in your great photo (sigh again) just meant the incoming IR was enough to balance the temp. Oh I'm sure one cloud would have had me diving for my parka, but as it was the marmots had it right, just basking in the sun. There were a family or two around my site and as long as I kept an eye on my food they were good neighbors.

this is a great help all: it's been far too long since I've come over the east side. Especially for someone as addicted as I to the high, stark and stoney that the crest offers in abundance.