Hi,
Yes, for that lake you can start at either Agnew Meadows or at the June Lake Loop (the latter is called the Rush Creek Trailhead).
Agnew Meadows is the prettier place to start and the prettier hike. But it's also higher (9000 feet?), and requires driving over the pass from Mammoth, which is at maybe 9600?
One year, we drove my non-4WD car to Agnew Meadows and hiked into 1000 Island and environs for a 3-day. It was the first week of October. The 3-day forecast was for 10% chance of rain.
On night 2, we got several inches of snow. Hiked out in a continued snowstorm. We lost the poorly maintained trail that goes downhill out of Garnet Lake towards the River Trail and our car. Lots of x-country on slippery snow on sloping granite.
Then, we returned to Agnew Meadows only to find we had to drive the very slippery and narrow road back to Mammoth in my non-equipped car. THAT was harrowing. And we were lucky the road had been plowed at all. Apparently, hours and hours went by with the town of Mammoth refusing to send a snowplow so that all the campers at Reds Meadow, and all the hikers like us, could get out. It only happened (after most of the day had gone by) after some muckety-muck stuck at Reds pulled some personal strings. We could have been stranded at Agnew Meadows for another night or two, next to our car. And since Tioga Pass was also abruptly closed, our trip back to the Bay Area got a lot longer as well.
After that I vowed never to risk going over that pass from Mammoth after Oct 1. You saw the part about how the forecast had been for 10% chance of precip? Well, the forecasts get notoriously unreliable right around now.
Personally I'd go for the Rush Creek trailhead, which is lower and easier to access.
– Elizabeth