I was up in Kings Canyon National Park about three weeks ago. They had the road to Cedar Grove plowed to the turn-off for Hume Lake. There was another road going up and left of the road (facing North) at about that same juncture. Looking on the map it seems to go out, connect with another road, and return a mile or so to the South. For the sake of planning though, I think you could just do an out and back. There was about two feet of snow on the ground at that point, enough to cover everything but keep that road flat (it was never plowed). There is also a place to park cleared on the side of the road. I would have gone snow camping out there if I had the time.
Other options in that area would be any cross country ski trail. There was one at the pass about 2 miles South of where I am describing, toward the entrance of the park (or about 8 miles North of the entrance) with good parking and some people milling around. I want to say it was Cherry ??? Ski trail.
I can't say how conditions are now, or even if the road is still plowed that far. Try to get a hold of the visitor center (I never got connected to a live person) and ask about cross country ski trails in Sequoia as well. They are usually fairly flat, are on or pass through flat terrain (good for camping), and have parking. They may even be groomed, which you want if it has become deep sierra snow.
The website has no useful information for Kings Canyon. So you'll need to speak to someone. Even the road that was gated and I skied on was plowed while I skied back, and it was still gated. You really won't need to go far out there. A couple hundred yards out and there won't be anybody else, especially in the winter.
Good luck, and bring a shovel.