Pratt and de Camp wrote another “ordinary guy falls into a fairy tale”, although maybe not as good as the Harold Shea stories, called Land of Unreason. It’s derived from pulp, and is, at the very least, a fast entertaining read.
The influential (and in my opinion, underwritten), Three Hearts and Three Lions has been mentioned, but Gene Wolfe’s The WizardKnight hasn’t. I saw some similarities-- guy (quite a young guy in this case) falls into a world sharing many features of Northern European myth, blunders around, and becomes a paladin. Not all that many people really like Gene Wolfe, but the ones who do like him a lot.
Varayan Memoir by Rick Shelley. Mostly serious, though the protagonist makes a lot of smart-aleck remarks, both spoken and mental. I remember it being pretty decent, but I read them around the time they came out, over 20 years ago.
I second the Darwath books. Hambly has a Master in medieval history, and her knowledge of the past informs her fantasy. Plus, she puts her characters through all kinds of hell. The first book The Time of the Dark was her first published book, and yet it doesn’t show. It’s very slightly dated in some language and attitudes, but it’s technically better written than some writers’ later books.
Dave Duncan’s protagonist in the Seventh Sword series doesn’t pass through a gate, etc. In fact, he’s not even in his own body (his new one is much nicer, actually) but he is an engineer from “our” world in a magical world. Literally magical, as one of the first beings he meets is a god, and the behavior of the River doesn’t make any sense without magic or miracles. But most of the action and situations are reality-based, to the point where Wally sometimes doubts any involvement by the gods.
Duncan’s Great Game series also has real-world magic-world crossover.