Since I have a bad habit (or perhaps a good habit, if you share my fandom) of bringing things back to Oz, I'll mention that items rendering someone invisible are fairly common in the series. The dama-fruit of the Valley of Voe makes anyone who eats it invisible, and Rosalie the witch has a ring of invisibility (hopefully without the One Ring's powers of corruption) in the Oz-related Sky Island. In Tin Woodman, the main characters come across a section of the Munchkin Country that makes everyone in it invisible. I believe the first appearance of a cloak of invisibility in the series, however, was not in Baum, but in Thompson's The Gnome King of Oz. The Flying Cloak of Invisibility, made by a sorcerer named Soob, not only renders its wearer invisible, but can also fly them anywhere they want to go. When Ruggedo and Peter Brown first find the cloak on board Polacky the Plunderer's old pirate ship, it's torn, and hence doesn't work. On the advice of his royal wizard Potaroo, Ruggedo takes the cloak to the Kingdom of Patch, where a Quilty...well, patches it. The Nome uses it to cause a lot of trouble after this, but he's done in by the fact that the blue patch remains visible. I tend to like stories that explain exactly how an imperfect magical item might malfunction, and this is no exception.
The idea of such a cloak is also incorporated into Jack Snow's Shaggy Man, but with a twist. In order to get past Glinda's invisible barrier around Oz (which Baum introduced in Emerald City as a way to end the series, and then largely forgot about when he returned to writing Oz), the King of the Fairy Beavers makes Cloaks of VISIBILITY.