By "Visitor Center", it sounds like you got info on the Schulman Grove facility. It is at the southern end of the White Mountain range, nearly 20 miles from the point you start hiking to White Mountain itself. It's that 20-mile section that you need to find out if/when it will open. It leads to a closed gate about 2 miles from the University of California's Barcroft Research Center, that you hike past when you climb White Mountain.
Here's a
Google map. You can see the pointer at Schulman grove. Drag the map down (south), and beyond the end of the white line (the dirt road), you will then see the "Barcroft Field Station". That's along your hike. If you zoom in on that map, it shows where the "locked gate" is where you would start your hike. Switch to Satellite view and zoom in more, and you can see the trail, etc.
You could also track down the UC offices in Bishop and ask them about the road. They are the ones who will be trying to get to their facilities asap.
