White Pocket is in an area of swirling twisted multicolored rock 5 miles due east of the Cottonwood Teepee trailhead. The White Pocket monolith can be seen from some of the high points at Cottonwood Teepees. White Pocket can be visited on it's own or as part of a trip to South Coyote Buttes. The road is a 4WD high clearance road due to soft powdery sand and sand ruts. Snow or rain may also make the road impassable. The drive took 50 minutes for 8.5 miles over frozen sand on a January trip. If you can make the drive to Cottonwood Teepees, the drive from there to White Pocket is just more of the same.
A BLM permit is not required for White Pocket so this trip is a good substitute if you can't obtain a permit for Coyote Buttes.
The hike from the cow pie filled parking lot to the rocks is only 0.2 mile on a wide sandy path. From there it's a matter of wandering through and over the rock formations without established routes. There's a fence north of the main "brain" rocks which you may have to work around. The Zion National Park Website's Paria Canyon - White Pocket page suggests hiking around the White Pocket monolith but we didn't have time for that. We spent an hour and a half in the area and that was sufficient to get a feel for the area but there's a lot more to explore. The Web site cautions about not breaking off fins, but I didn't see any like those at Coyote Buttes South: Cottonwood Teepees.
I suspect the White Pocket monolith is named for the pockets in the "brain" rock near the trailhead but haven't been able to confirm that. In any case, there are about a dozen pockets in the rock. They are off to your right at a 45 degree angle at the end of the path from the parking area.
You'll also notice an impressive double arch in the cliffs to the east. I took several photos of the arch including some with a 12X zoom. I didn't realize until cropping the photos at home, that it's not an arch but merely two shallow alcoves in the cliff.