While no other griffins appear in Baum's Oz books, the Nome King's Chief Steward does use the expression "cross as a griffin" in Ozma. Ruth Plumly Thompson coined a similar simile, "hungry as a hippogriff," in Lost King. And speaking of hippogriffs, their name might have been the inspiration for that of the Hip-po-gy-raf from Tin Woodman, even though the more obvious sources for this creature were the hippopotamus and giraffe.
But it's in Thompson's Jack Pumpkinhead that we finally see a griffin used as a major character. This would be Snif the Iffin, an inhabitant of the Land of the Barons in the Quadling Country. When he flew into Scare City, he was chained up by the ghoulish inhabitants for five years. During this time, he forgot his name, and took on the appellation of "Snif," which the Scares gave him. He also lost his growl, and decided that a griffin minus the "grrr" was an Iffin. This title fit quite well anyway, due to Snif's philosophical nature, and he's usually referred to as "the Iffin" even after he regains his growl. The Iffin is a powerful but gentle animal, subsisting on flowers and frequently making up verses (which I guess would make him a flower-eating poet). Snif also looks somewhat different from your typical griffin. According to Thompson, "It looked not unlike pictures [Peter] had seen of this rare and fabulous monster--being sandy red in color, with a huge lion's body and dragon's claws. Its head, instead of being the usual eagle head, was of rather a doggish nature with a stand-up mane and inquisitive, pointed ears." Since Neill also gave him the scaly wings and barbed tail that he gave Mombi in griffin form, the Iffin wound up looking not much like a lion or an eagle at all, but more of a cross between a dog and a small dragon.
As with many of Thompson's original protagonists, Snif moves into the Emerald City at the end of his starring role, but is mentioned only in passing after that. Perhaps his main claim to fame in later books is that he's briefly turned into a china cat by Ruggedo in Pirates. (This happens alongside the former Nome King's transformation of the Cowardly Lion into an iron dog. I have to wonder if Thompson accidentally got these two backwards, since it's the Lion who's a big cat, while the Iffin has canine features.) Later on, however, Phyllis Ann Karr wrote a story called Maybe the Miffin, in which the Iffin searches for the titular character, whom he believes to be his female counterpart. The character has been a favorite of mine for some time, and it's unfortunate (for the fans, that is) that he's still under copyright.