Hungry Tiger not only gives us the Evian desert dominion of Rash (described in some detail here), but another desert area within Oz itself, where the winding road deposits Betsy Bobbin and Carter Green after dragging them around the land. This area has a patch of quicksand, and is where the Quick Sandals are kept. It's in Yellow Knight that we next see another significant desert country in Oz, however. This is Samandra, located in the small stretch of Winkie Country to the east of Oogaboo. Samandra is a prosperous and well-established country, with some of its inhabitants being over 700 years old, but it is perhaps best known as the only part of Oz in which animals are unable to talk. It was also the original home of the Comfortable Camel and Doubtful Dromedary. As far as its history goes, the Sultan of Samandra allegedly feared an alliance between his two neighboring countries of Corumbia and Corabia if the prince of the former married the princess of the latter, so he transformed the Corumbians into trees, and the Corabians into fishes and frogs. They remained this way until the events of Yellow Knight, and after they were restored, Ozma stripped the Sultan of his sorcery.
Wishing Horse once again uses a desert monarch as its main villain, but at least this time he's somewhat more sympathetic. King Skamperoo of Skampavia, a country adjoining Noland, tires of his dull and barren homeland, and uses some magic that he comes across to declare himself Emperor of Oz. He is eventually persuaded to return to Skampavia, only this time he has more sense, a talking horse to advise him, a greater willingness to listen to his prime minister, and some magic left to improve the climate of his own country.
I suppose it's also worth mentioning Enchanted Island, which not only mentions the Desert of Ho-Taro as one of the neighbors of the wool-producing Gillikin kingdom of Kapurta, but also that the camel Humpty Bumpty was originally the favorite steed of the Shah of Hah Hoh Humbad. This place is described as a lazy desert city in the Munchkin Country, not too far from another city called Drumbad. It seems that many of these Ozian desert countries are located along the edges of the Deadly Desert, which would make sense. Kabumpo even comments in Silver Princess that "[a]ll the countries bordering on the Deadly Desert are queer no-count little places." This isn't entirely true, however, as the quite pleasant Jinxland also borders on the desert. It doesn't appear that the destroying sands have the power to enter Oz, but they can perhaps affect some (but not all) of the nearby climate.