I agree with Jonathan Mayhew that
The New Critics developed theories sympathetic to some aspects of literary modernism, but they condescended to Pound, Williams, and Cummings, tolerated Moore, ignored H.D., did a poor job with Stevens, failed to welcome O'Hara and Creeley, Duncan and Ginsberg--the poets who learned from modernism. Modernism, for the New Critics, didn't include European surrealism or the Latin American offspring of the avant-garde.
My own approach to literature and literary criticism is strongly informed by the example of New Criticism, although substituting for the New Critics' tendency to treat the text as object an emphasis on aesthetic experience, inspired by John Dewey's Art as Experience. The New Critics' insistence that the proper focus of literary study was on literature, not politics or sociology, that reading requires paying close attention to the organization of language, and that whatever "meaning" a text conveys is necessarily conditioned by its formal organization have always seemed to me not just convincing but finally so manifestly obvious I can't really accept the judgment implicitly rendered by the practice of academic criticism that the underlying assumptions of New Criticism are inextricably tied to a historical phase of literary criticism that was appropriately superseded by the new assumptions of subsequent phases. The basic principles illustrated in a book such as Cleanth Brooks's The Well-Wrought Urn are still valid, although the notion of "well-wrought" is more useful than the objectification inherent in the metaphor of "urn."
But it is true that while it could be argued that New Criticism emerged from the theory and practice of modernism, many writers who were either certified modernists or who were influenced by the innovations of modernism did not find favor with most New Critics. The innovations of Eliot and Faulkner were more congenial to the conservative formalism (and ultimately more conservative cultural outlook) of the major--mostly Southern--New Critics than William Carlos Williams or E.E. Cummings. One imagines that those of these critics who survived through the 1960s and 1970s found postmodernism pretty distasteful, although luckily they chose not to pronounce on it much, and surely Language poetry was/would have been anathema to them. I would conclude that in general most "avant-garde" writing seemed to the New Critics a repudiation of form, or at least a failure to regard it with the care they found in the greatest poetry, Modernist and traditional. So if in some ways New Criticism was a product of the increasing self-consciousness about form to be found in modernist poetry and fiction (and especially a further refinement of the early essays of T.S. Eliot), by no means was all experiment in form, or every writer who emphasized form, sanctioned by New Critics.
This was more than short-sightedness on these critics' part. It exemplifies a real blindness to some of the important implications of their own critical principles. In my view, a formalist approach to literature ought to value most of all those texts that do experiment with form, those writers who seem to foreground formal experiment for its own sake. Only such experiments keep literary form from ossifying into an empty formalism and reading from degenerating into a formulaic exercise. Not all of these experiments will be successful; probably only rarely will an experimental work truly reward the kind of alert, dynamic reading the New Critics at their best both championed and exemplified, but then only rarely does any work of fiction or poetry so reward our attention. Experiments in form keep "form" as a defining characteristic of literary art alive as a subject of critical commentary and as an object of readers' consideration. No doubt the New Critics' religious and cultural commitments kept them from fully appreciating the value of uninhibited experiment in literature, as it seemed to them implicitly subversive of the final order they valued more highly.
As stated most directly by T.S. Eliot in "Tradition and the Individual Talent," the New Critics could accept the challenge to order presented by the "truly new" work of literature, but they conceived of the new as both changing and adapting itself to the established order (of literary history, at least). I agree with them that the new in literature works in this way, but see no reason to think the principle doesn't ultimately work with even the most radical departures from convention. Until such time as innovations in poetry and prose literally cross boundaries and create something that can no longer be classified as either, or until writing itself no longer attracts new talent, "avant-garde" works will continue to offer up fresh responses to the demands of form, which will in turn ultimately be assimilated into the tradition, however different that tradition will be after such efforts have been registered. I would say that the formal audacity of Eliot's own work (as opposed to its expression of alienation from modern culture, which has indeed become its own kind of cliche) has yet to be altogether assimilated, and will continue to play a role in inspiring new avant-gardes.