Of course, Thompson was not the first author to feature underground lands in Oz books. Dorothy and the Wizard had an earthquake bring the title characters to the Vegetable Kingdom, inhabited by the heartless Mangaboos. They live in a city of glass lit by a large sun (well, relatively speaking, that is) surrounded by five smaller ones. As the country is close to the center of the Earth, gravity is weaker there, and it's possible to walk in the air. When the characters escape from the Mangaboos' land, they journey through the verdant Valley of Voe and the silent wooden Land of Naught before almost reaching the surface. Getting back to Thompson, she had the East-Asian-styled Silver Island at the bottom of the Scarecrow's beanpole. While obviously inspired by the idea of getting to China by going through the Earth, the fact that the beanpole grows out of the ceiling of the silver palace rather than the floor suggests that the island is underground. The Golden Islands lie nearby, but the rest of the geography of this region is never explored. The Silver Islanders themselves describe themselves as older race than the Chinese, being people of the stars, rather than the sun. (And yes, the Silver Island does have both sun and stars above it, despite its location.) Grampa features a visit to Fire Island, located in a sea of boiling water, inhabited by people made of flame, and ruled by a man of molten iron. There is also a mainland nearby, where the giant Blazes controls a volcano with its top in Ev.
Of course, the most frequently visited underground land in the series is the Nome Kingdom, which is seen in nine canonical books (Ozma, Emerald City, Tik-Tok, Rinkitink, Hungry Tiger, Gnome King, Wishing Horse, Lucky Bucky, and very briefly in Pirates), with Nomes (especially Ruggedo) appearing in several others as well. The extent of the Nome Kingdom is never clear, but it seems immense.
I have to suspect that Thompson's nine levels of the Underworld are based on Dante's nine circles of Hell. The Ozian Underworld isn't the realm of the dead, but there are some similarities there. The plots of Ozma, Tik-Tok, and Rinkitink all involve descending into the Underworld (specifically the Nome Kingdom) in order to rescue someone, bringing to mind the myth of Orpheus. The Nome Kingdom, like Hades, is also a very rich country, full of precious minerals. Some of the underground countries are loaded with monsters, including invisible bears, a naked wrestling giant, ravenous fire-fish, and blind dragons. And the Nome Kingdom is said to have seven-headed dogs, obviously inspired by Cerberus. I suppose such connections are more or less inevitable when dealing with underground fantasy lands.