Park up overnight in town and you will get attention. There is plenty BLM land around that you could overnight on, if you can get to it - we have had a lot of snow recently and more on the way. Roads and parking lots are best to be clear of vehicles to ensure hassle free snow plough action.
The safer bet is to head down towards bishop and use one of the campgrounds down there. It only adds 40 mins at the start of the day, and hassle free.
There's a great area about 10-15 minutes north of Bishop on the west side of the Owens River. Go like your going to climb in the gorge and keep taking the approach road north past the parking areas, eventually it turns to dirt roads and there's tons of camping (no idea what the conditions are there now). Puts you 45 minutes from Mammoth, but there's likely closer spots