Covered Bridge or Blackberry, on the eastern end of the Kanc, are much closer (stay on West Side Rd, DON'T take rte 16 through Conway). They are first-come-first-served and practically guaranteed to have room this time of year (they stay open through October).
Numbers 8 and 4 on this map:
http://www.fs.fed.us/r9/forests/white_m ... d_map.htmlBut you should be using the AMC hiking maps for this kind of planning (in the White Mountain Guide or sold separately -
http://www.amazon.com/AMC-White-Mountai ... 1934028045 Not detailed enough for off-trail navigation, but great if you're wondering where trailheads and campsites can be found, or whether you can use Dugway Rd as a shortcut from Covered Bridge campground to West Side Rd (you can, but it's usually not worth it).)
PS Make sure you give Whitehorse a full day to dry off after a rain, or stick to a route that you can see all of from below.