If you want few people.... don't do Thousand Island. People dayhike and thruhike that area constantly. Pack trains deliver gear and food for folks. It's a zoo along the John Muir Trail.

But the scenery is hands down the most stunning you'll find...

for less traveled areas (tho not totally untravelled, they are popular to locals like me) look at trails around Dinkey Lakes, Ansel Adams, John Muir wildernesses from the southerly approach rather than coming out of Yosemite or up from agnew/reds. The scenery there is classic Sierra, granite and lakes and meadows and flowers blooming as late as September.

Golden Trout will be lower elevation but good fishing.

Emigrant Wilderness off highway 108 is hammered by stock, but also gorgeous, and bordering northwestern Yosemite.

Sequoia/Kings will be easier to get backpacking permits for than Yosemite and there are classic Sierra hikes to do, the hike to Alta Peak is hands down my preference over any of the Yosemite peaks, and while you will not be alone there will be maybe a dozen folks at most on top. Nothing like the hundreds who hit Half Dome, or the dozens you will find on the trail near Thousand Island.

Rae Lakes loop - absolutely gorgeous, need a week to really enjoy it. Again, you won't be alone. But not nearly as jam packed as other trips.

As for the snow... it will be melting fast, which makes creek crossings dangerous in some places. By early July we will be heading up to around 7,000 feet - I am expecting to have no issues with snow until 9,000 or more. But that's just my estimate. I was up in 9,000+ in Sequoia earlier this week and the drifts were deep... in Yosemite we hit small patches at 6,000 and found lots of it at 7,000. But a month with higher temps could see that diminished a whole lot. Keep checking.