White Mtn. is a long way from winter's road end. My guess it is best done as a ski tour during those months.
I've been up to Schulman Grove in May a few times, when you can drive as far as Sierra View, ~9,800' to 9,900'. We snowshoed the grove and explored the mine adits, which are visible from the road. I do remember someone trying read the peak utilizing snowshoes and a sled, failing both times he tried it.
San Jacinto is fairly easy as long as there isn't a blizzard howling. A blizzard makes it more interesting and fun, though. San Gorgonio is a bit of challenge requiring climbing gear from both sides.
