In 1926, Ruth Plumly Thompson published The Curious Cruise of Captain Santa. About half the size of one of her Oz books, it was illustrated by the wonderful John R. Neill.
Whether or not Thompson was aware of Baum's The Life & Adventures of Santa Claus is no issue, as Thompson did not tie her take on Santa Claus to Oz at all. Santa lives at the North Pole, with a boy he took in named Jimmy Christmas, a penguin named Penny (never mind that penguins live in the South Pole), the brownies who make the toys, and they have a friend at Snowshoe Mountain, Huggerumbo, a polar bear.
Santa builds a ship called the Chimneypot and with Hugger, Jim, and Penny, sets sail to tropical islands to find new Christmas presents, and they even go beyond the sunset, where there are live toys.
And honestly, that's the plot.
I just want to say that I don't feel this book should be counted in the same continuity as the Oz books. Thompson doesn't mention Oz, and her Santa is very different from Baum's Claus. Thompson's Santa is lively, spontaneous, and a little light-headed. Baum's Claus is wise, solemn, and quiet. While both Santa and Claus are fun to read about and both are excellent characters, I think Baum's Claus is the Santa Claus for Oz, as Baum, who created Oz, directly tied his version with Oz. Thompson never did, although John R. Neill had a barrel-bird (who help Santa load the Chimneypot) appear in Lucky Bucky in Oz. Whether barrel-birds also exist in Oz or somehow it crossed over is for individual readers to decide.
But all the same, I did enjoy Thompson's book, and can suggest that Oz fans would enjoy it. It's available in an inexpensive paperback from the International Wizard of Oz Club.
Next time, Jack Snow's short Santa story.