I grew up hiking and backpacking around the June Lake Loop. We spent many a summer (and sometimes a fall and winter) tramping through the area. Not many lakes and passes that I haven't seen through there.
It looks like recent snows are sticking in the higher elevations now. That will probably rule out a lot of things past Gem Lake above Silver. The best hikes in that area (in my humble opinion, anyway) are in the Ansel Adams Wilderness between the Rush Creek trailhead at Silver Lake, and Agnew Meadows in Mammoth. My favorite trips usually begin at Agnew Meadows, and end at Silver Lake, with any number of lakes and mountains visited in between, dependent on how much time I have.
Steves recommendation of Fern Lake is solid, but it's a short day. It's not very long distance wise, but it is fairly steep. And gorgeous. It's a pretty little lake. The Yost Lake trail will take you past Fern Lake up to Yost Lake, and that's more of a day trip. Yost isn't as pretty as Fern in my opinion, but still very worth seeing. I doubt these areas are as snow covered as what lies above the Agnew dam at this point.
The Yost Lake trailhead has parking, but when I was up there this summer, the parking area had been closed to motor vehicles all three times. Not sure if it's open at this point or not. There's plenty of places to park nearby though, it shouldn't be a problem.
