that would be a 1.5hr one way drive from Lone Pine to Big Pine and up rte 168 and turn off to White Mt Rd. I've been to the Sierra Overlook there many times for a glorious morning view of the Owens Valley and the looming Wall of the Eastern Sierra. 2 miles beyond that is the Schulman visitors center/short discovery hike of 30 minute or so, some old trees, and thin air (thin enough for day one!) hint, hint at 10,000 ft.
12 miles on dirt road beyond that will get you to the oldest trees ( Patriarch Grove). beyond that is the jeep trail to White Mt. I have not done that. Here is one persons report
White Mt TR That side is very dry, barren and long. Personally, I never had any enthusiasm for going there. An alternative? Much prettier, say, going to Bishop South Lake, dayhike from there, maybe few miles round trip to Treasure Lakes 10,700 ft, for example. No need to wear yourself out trying to acclimatize on the long, hot, dry, shadeless 14,000 ft White Mt .