The Trail of Cthulhu

Derleth and Lovecraft corresponded, to the extent that Lovecraft created the character the Comte d'Erlette (author of the fictional "Cultes des Goules") in his stories.
Mostly building on the Innsmouth tales, although one also references the story Lovecraft wrote for Harry Houdini as the team head into the desert looking for clues.
The stories all mesh quite well, and Derleth does a fair imitation of the Lovecraft style. It would probably be classed as fan fiction these days, adding some elements to the Cthulhu Mythos, and trying to bind some of the Old Ones/Elder beings into a pantheon of warring gods.