White Mountain is a good distance from Lone Pine. Google Maps shows 82 miles. Getting to Big Pine and up to Westgard Pass is a nice drive, and so is the paved section of White Mountain road. But then there is a 20 mile section that takes at least an hour each way. So one way, it is 2.5 to 3 hours.
The nice thing is that you start hiking at 12,000 ft elevation, so the hike is great for acclimating. It would be good for two days before Whitney, but not the day before, since it is still pretty strenuous -- 14 miles round trip.
For an easier day before Whitney, you could just drive to the first Bristlecone Pine forest on that road, Schulman Grove. That would cut almost an hour off the drive, all on paved road, and gets you to 10,000 ft. And there is a nice loop trail around the grove so you could get in a little hike.
Here's a map of that area:
postholer.com / Schulman GroveEdit: After posting this, I see + @ti2d beat me to the same suggestion.
