Huh, one thing that strikes me as postmodern in a way that isn;t trying to be postmodern is the quirky film.
Just watched The Hunt for the Wilderpeople which I would describe as quirky, alongside such films as those of Wes Anderson, Michel Condry, Jean-Pierre Jeunet, etc (those three are among my favorites).
There was actually an essay I read a while back about quirkiness as a genre - forget if it was theory-based or not - and it went into detail about characteristics of these films. A sense of humor that is austerely incongruent or outright absurd, some of the characters being weirdos in a glorified manner, etc.
I could be overlooking some earlier films, but quirky does seem to be a somewhat new category. There are certainly films with quirky protagonists, but quirky as a genre seems to be an artifact of the indie film industry, however paradoxical that phrase may seem.
I'm thinking this picks up maybe around the early 70's after the whole easy rider raging bull wave, perhaps paving the market for quirk as a subgenre of counterculture?
Hints of it in Linklater, Smith, Tarantino, etc.