There was only one L. Frank Baum, and in 1921, likely no one knew this better than Ruth Plumly Thompson, his appointed successor. She'd written a couple children's books and had written short children's pieces for newspapers. She was also a fan of Baum's series, and was doubtless overwhelmed when she was approached to continue the series.
So, what was she to do? She took her own spin on Oz, and tried to respect Baum's original as much as she could without stifling her own creativity.
A good friend of mine sent me two of Thompson's books, The Curious Cruise of Captain Santa and Speedy in Oz. Really, the only issues I have with Thompson's books (continuity-wise), is that she killed Mombi (albeit "offscreen"), did away with the first Nome King (though that's been written around), gave the Scarecrow an origin that I don't agree with, and re-wrote the life of the Good Witch of the North (also been written around, and "Who's Who in Oz" seems to suggest that Jack Snow was going to do likewise).
Now, Speedy in Oz, I didn't have a real problem with, probably because most of the book deals with little of Baum's creations, except that it's set in the world he created.
One thing to remember is that if Baum wrote solely to entertain, Thompson did it even more for that purpose. While I prefer Baum's writing, Thompson is entertaining and fun, and manages to tell a story that will keep the reader interested.
The story opens with Umbrella Island, an island kept afloat in the sky by a giant magic umbrella, hitting Loxo the Giant in the head. He is so angry that he demands that in return, he be given a child of his choice to lace to his boots for him, and picks out a "boy" close to the king, who is named Sizzeroo. The people manage to convince Loxo to give them three months to prepare the "boy" for his future life, but it was actually Princess Gureeda who was selected. (The same style of clothing is worn by all inhabitants, which John R. Neill draws as resembling pajamas, though Thompson compares them to stereotypical Chinese garb; Gureeda also wears a single braid in her hair, like men on Umbrella Island usually do.)
Over in the Great Outside World, Speedy, a former visitor to Oz, is helping his Uncle Billy in his work at an excavation dig, when they get to a complete dinosaur skeleton. Speedy convinces his uncle to put it together, but shortly afterwards, a gyser erupts, carrying Speedy and the skeleton high into the air. At the same time, the dinosaur's bones are fused together, and it comes to life. Able to speak English, Speedy names it Terrybubble (when he mispronounces "terrible"), and Terrybubble manages to keep Speedy in his chest, and they land on Umbrella Island.
When one the King's advisers sees Speedy, he suggests that Speedy be given to Loxo instead of Gureeda, but Speedy finds a friend in Waddy, the Wizard of Umbrella Island, and soon, everything comes to a head, with Ozma, the Wizard, and the Scarecrow on the sidelines as the climax is reached.
I noted that, unlike Baum, Thompson did not have the characters visit strange countries that did not bear on the plot. (There is a misadventure with some warring islands, but the result of this becomes a bit of a key element in the rest of the plot.)
I really liked Speedy in Oz and hope to get some more Thompson Oz books soon! (Only five more days.... And it's mine!)