Perhaps its predecessors can be found in the now vast catalogs of cyberpunk, or its more nebulous younger sister, steampunk, but for a time the deliberately subversive and cliche-defiant umbrella term "New Weird" subsisted as a signal that something new was happening in speculative fiction. For readers fatigued from the same old space operas and Tolkein cookie-cutter knockoffs, authors such as Jeff Vandermeer, China Mieville, Lucius Shepard, Michael Moorcock, and the "biopunk" Peter Watts (among others) reinvigorated the landscape (er, or at least the bookshelf).
That said, start with the anthology The New Weird and see what you think. Other good starting points would be the Bas-Lag series by Mieville, and Vandermeer's inventive City of Saints and Madmen.
Thinking of the New Weird, by necessity there must've been an Old Weird as well-- and indeed, "Weird Tales" authors like Robert Howard, H.P. Lovecraft and others such as Ambrose Bierce and Algernon Blackwood remain something of a launching pad for any standard of "weirdness" to be possible in our day.
Then again, Georges Bataille, Thomas Ligotti, William S. Burroughs, Angela Carter, Jorge Luis Borges, and Reza Negarestani clearly have zero "anxiety of influence" when it comes to surreal storytelling, horrific realities, fantastical monstrosities, and imaginative absurdities.
Not to be reductive and lionize such a motley inventory of people simply for their textual transgressions (the Super-ego's 'But-but-but....you can't write that!'), the compelling ingredient in "Weird" or speculative fiction reveals itself only as it writes itself: if genre is the surefire algorithm of publishers to attain fiscal success and wrangle some order on human imagination, then "Weirdness" realizes itself as the genre of no-genre.
Such a dynamic is best thought of as less paradox than it is simply a dialectic: wherever the 'latest' trend is trumpeted and ossified that a bookstore (or even a library!) can shelve it, then that is precisely the signal the trend is dead, or moved on to new realms.
Consider this post then a eulogy then. Long live the Weird.