A number of horse-accessible trails meander through Sequoia and Kings Canyon [1]. Rent a horse for a day ride from the pack stations at Cedar Grove (559/565-3463 summer, 559/337-2314 off-season) and Grant Grove (559/335-9292 summer, 559/337-2314 off-season). If you’re new to horses, pick a gentle one-hour ride. For more hardy or experienced riders, both pack stations offer half-day and full-day rides.
If you’re looking for a multi-day horseback adventure, you can find that too at Sequoia and Kings Canyon [1]. Call either of the pack stations to arrange a guided backcountry pack trip.
Links:
[1] http://www.moon.com/destinations/california/yosemite-and-the-eastern-sierra/sequoia-and-kings-canyon-national-parks