The most difficult aspect of this route description will be the drive up FR278 on the west side of Mt. Antero. From the intersection of FR277 & FR278, head SSE on 278 continuing up through forest until you reach the last trees at just above 12,000 ft. The road turns east and heads briefly up a gully before making the first of six sharp switchbacks to gain elevation on the flank of Antero. After the sixth switchback, the road heads generally south and crosses a ridge at 13,100 ft. This switchback section of the road is the roughest and most tedious part of the drive (or hike). If hiking this section, attempts to shortcut the switchbacks will lead to frustrating loose rock conditions on a very steep mountainside.
At the ridge crossing at 13,100 ft., there is a fork in the road. Going left grants access to Antero and Mount White. Drive/walk a few hundred yards east, then take a right fork that drops from 13,100 feet to about 12,850 ft. This road forks cuts down across a tundra expanse. Part somewhere in the vicinity of the Antero/White saddle if driving.
The trip up to White's summit is straightforward. There's an old mining road that heads SSW, gaining elevation across tundra to White's SW ridge. The same road switchbacks and follows the ridge east, but ends at some mining claims at 13,300 ft. Continue following the ridge east to the summit over the whitish rubble/talus and boulder rubble at times. You can also just skip the road and walk SE more directly to the SW ridge over tundra, embedded rock & scree. There are two apparent summits about 400 feet apart. As simple as this peak is, you might as well tag both just to make sure. On Google Earth we obtain a reading of 13,657 for the west summit and 13,664 for the east, but we do not find GE to be very accurate at all. The summit will afford nice views looking down into Browns Creek and across the valley to Shavano and Tabeguache.
If looking for a way to make Mt. White a more sporting ascent, it can be accessed from the Browns Creek trailhead, east of the summit by several miles. The round-trip hike will gain about 4,800 feet and total 13 - 14 miles in length. If hiking from a suggested vehicle park at 12,000 ft. on FR278, the one-way mileage will be 4.0 miles and 1,950 in elevation gain.
Links to other information, routes & trip reports for this peak that may be helpful.