Fritz Leiber!
A few years ago there were several collections of his work available, if a bit pricey (Midnight House editions). Not sure what you'll find now. Among his most anthologized stories, so you may find them more easily, see "Smoke Ghost," "The Girl with the Hungry Eyes," "The Dreams of Albert Moreland," "Diary in the Snow." (You might even find a couple of these on-line.) I'm particularly fond of the novella "You're All Alone" though less so of that story at novel length, The Sinful Ones. "Four Ghosts in Hamlet" is a delightful, even funny horror story. Old pbs of Night's Black Agents may be findable and are a good introduction. His novels Conjure Wife and Our Lady of Darkness are also quite good, and short.
I'd also suggest The Best of Richard Mathson and Perchance to Dream by Charles Beaumont; both were writers for the original Twilight Zone TV show and some of the stories in these collections were adapted for the show, and most display that sort of imagination. Beaumont, unfortunately, died young. About Matheson, his I am Legend is both an s.f. and a horror classic, influencing Night of the Living Dead, and his impact on popular culture in the late 20th century is hard to underestimate. As writer or screen writer he was involved with "Duel," "Nightmare at 20,000 Feet," The Legend of Hell House (based on Hell House), The Nightstalker, Trilogy of Terror, several Poe-based movies from Roger Corman. Stephen King has called Matheson his greatest influence. Haunted Castles by Ray Russell, like the Matheson and Beaumont, comes from Penguin Publishing and is still in print; it's quite good if you like Gothic story-telling and don't mind a little 1960's sexual politics.
Another one that seems obvious but could be overlooked easy enough, The October Country by Ray Bradbury. For me, Bradbury had such a feel-good public persona it's easy to forget he could write disturbing fiction.
A good source of short fiction is The Dark Descent, though it does have stories on either side of that time range. The follow up volume, Foundations of Fear is also recommended.
Names to look for,
Carl Jacobi
Elizabeth Bowen (literary writer who wrote a good number of ghost stories)
Fredric Brown
Robert Aickman (if you like ambiguity, you'll love Aickman; "The Hospice" and "The Inner Room," for example, are excellent, as are other stories by him)
Shirley Jackson
Daphne du Maurier
Henry Kuttner & C. L. Moore (after marriage much of their writing was done together, some of it, including s.f., was also horror)
Theodore Sturgeon (known for s.f., still "Killdozer," "It," "The Professor's Teddy Bear" and others are horror and influenced other writers like Bradbury)
Joseph Payne Brennan (I'm iffy about JPB, but others enjoy him more and "The Green Parrot," "Slime," and "Canavan's Back Yard" all have qualities to recommend them)
Manly Wade Wellman
Eric Frank Russell (another Midnight House edition collected all or most in one volume; good fun in small doses)
John Collier (Fancies and Goodnights is available through New York Review of Books editions; a mix of horror, fantasy, mystery, I can't recommend this volume highly enough)
Gerald Kersh (see Valancourt Books)
Some of the stories of L. P. Hartley (The Traveling Grave has recently been reissued from Valancourt Books) and Walter de la Mare were written and published before 1939, some after, much of it is worth a look, especially Hartley's "A Visitor from Down Under" and de la Mare's "Seaton's Aunt".
I'll probably think of more as soon as I hit "Post reply" but I'm sure others will join the fun shortly, too, with their suggestions.
Randy M.