There are many reasons to think that Eric Havelock had crucial influence on McLuhan long before the publication of Havelock’s Preface to Plato in 1963:

• Havelock’s orality research was well known at Toronto when McLuhan arrived in 1946.¹
• By 1947 at the latest the same was true at Harvard (after Havelock’s guest lectureship there in the 1946-1947 academic year and his joining the faculty in 1947). In the fall of that year I.A. Richards, McLuhan’s old professor from Cambridge and at Harvard since 1939, described Havelock’s orality research in a BBC talk. A transcript appeared in the BBC Listener and McLuhan may well have seen it since news of it must have been abroad at UT among Havelock’s many friends and former colleagues there.²
• In 1946-1947 Havelock published Virgil’s Road to Xanadu in three parts in the new UT journal Phoenix.  This essay details a view of poetic composition that McLuhan would explore in depth over the next decade and that led him to his central findings (a relativity theory of human experience and the roll of co-variable senses in demarcating different types of experience within that theory).³
• Havelock and McLuhan appeared in the same issue of UTQ in 1948 where Havelock’s review specifically raised the question of oral vs literate composition.⁴
• McLuhan’s 1954 lecture, ‘Eliot and The Manichean Myth As Poetry’, mentions Havelock’s 1951 Crucifixion book: “Today many thoughtful people are torn between the claims of time and space, and speak even of The Crucifixion of Intellectual Man as he is mentally torn in these opposite directions.”⁵

Aside from these markers in which Havelock is explicitly involved, it may be that McLuhan’s references to the work of Milman Parry and Albert Lord point in the same direction.⁶ For aside from a  footnote to an obscure paper from Parry⁷ in Innis’ Empire and Communications (itself doubtless owing to Havelock), Havelock would seem to have been the source of the immense importance attributed by McLuhan to the work of Parry and Lord on oral composition and performance.

In fact, McLuhan begins The Gutenberg Galaxy (1962) by situating it in relation to Parry (who died in 1935, age only 33) and Lord (who was a colleague of Havelock at Harvard during Havelock’s time there from 1946 to 1961).  His ‘Prologue’ begins:

The present volume is in many respects complementary to The Singer of Tales [1960] by Albert B. Lord. Professor Lord has continued the work of Milman Parry, whose Homeric studies had led him to consider how oral and written poetry naturally followed diverse patterns and functions. Convinced that the poems of Homer were oral compositions, Parry “set himself the task of proving incontrovertibly if it were possible, the oral character of the poems, and to that end he turned to the study of the Yugoslav epics.” His study of these modern epics was, he explained, “to fix with exactness the form of oral story poetry … Its method was to observe singers working in a thriving tradition of unlettered song and see how the form of their songs hangs upon their having to learn and practice their art without reading and writing.” [The Singer of Tales, p. 3] Professor Lord’s book, like the studies of Milman Parry, is quite natural and appropriate to our electric age, as The Gutenberg Galaxy may help to explain. (…) The enterprise which Milman Parry undertook with reference to the contrasted forms of oral and written poetry is here extended to the forms of thought and the organization of experience in society and politics. That such a study of the divergent nature of oral and written social organization has not been carried out by historians long ago is rather hard to explain. Perhaps the reason for the omission is simply that the job could only be done when the two conflicting forms of written and oral experience were once again co-existent as they are today. Professor Harry Levin⁸ indicates as much in his preface to Professor Lord’s The Singer of Tales (p. xiii): “The term ‘literature’, presupposing the use of letters, assumes that verbal works of imagination are transmitted by means of writing and reading. The expression ‘oral literature’ is obviously a contradiction in terms. Yet we live at a time when literacy itself has become so diluted that it can scarcely be invoked as an esthetic criterion. The Word as spoken or sung, together with a visual image of the speaker or singer, has meanwhile been regaining its hold through electrical engineering. A culture based upon the printed book, which has prevailed from the Renaissance until lately, has bequeathed to us — along with its immeasurable riches — snobberies which ought to be cast aside. We ought to take a fresh look at tradition, considered not as the inert acceptance of a fossilized corpus of themes and conventions, but as an organic habit of re-creating what has been received and is handed on.” (…) Such a reverse perspective of the literate Western world is the one afforded to the reader of Albert Lord’s Singer of Tales. But we also live in an electric or post-literate time when the jazz musician uses all the techniques of oral poetry. Empathic identification with all the oral modes is not difficult in our century. (…) The work of Milman Parry and Professor Albert Lord was directed to observing the entire poetic process under oral conditions, and in contrasting that result with the poetic process which we under written conditions, assume as “normal.” Parry and Lord, that is, studied the poetic organism  [before]⁹ the auditory function was suppressed by literacy. They might also have considered the [contrasting] effect on the organism when the visual function of language was given extraordinary extension and power by literacy. (The Gutenberg Galaxy, 1-2, 3)

After the 1963 appearance of Preface to Plato, McLuhan continued to cite the work of Parry and Lord, usually along with Havelock, as follows:

The invention of script provided a technology that created extensive new environments. The content of script was at first the oral tradition of poetry and wisdom. Just how the content of script was affected by the new medium of writing is a story that has been told by Albert Lord in his Singer of Tales and by Eric Havelock in his Preface to Plato. The new technology, in creating new environment for the old technology, maximizes change. Yet the environmental is also the unnoticeable. We seem to be least conscious of the most archetypal technologies. (New Media and the Arts, 1964)

Eric Havelock in Preface to Plato (…) explains the relation of Homer to the “tribal encyclopedia”. In presenting the bard as the traditional educator of Greeks, Havelock indicates that poetic performance was a kind of group ritual that involved the public like any jazz or rock festival. As Parry and Lord have indicated in The Singer of Tales, in an oral culture performance is also composition. Havelock devotes much of his book to explaining the effect of the phonetic alphabet on the Greek sensibilities and culture, pointing to the rise of self awareness and identity as an immediate result of this unique form of codifying experience. What especially needs to be noted about the phonetic alphabet in our time is its power to impose its assumptions on wide fields of operation and experience. (Reading and the Future of Private Identity, 1973)

The source of inspiration and the group who are to be helped or enlightened are one and the same in an oral culture. We are all familiar with the work called The Singer of Tales, about the conditions of oral composition in the Yugoslav epic. Parry and Lord, the authors of this book, studied the conditions of oral composition, those of Homer but also those of the modern jazz musician. Even in the popular art of jazz, the public is the immediate participant in the composition. There is no written score; there is a vast store of formulas which are used according to the needs of the moment and the particular occasion. Improvision is the mode of composition. It is only by improvising that the public can participate in Art, in the creative activity of Art. This event, as big an upheaval in human affairs as ever occurred, puts the Third World directly in the centre of the picture as the kind of world we are now beginning to share everywhere, the kind of world we need to understand in order to understand ourselves. I think we should concentrate on that theme: Third World as centre of the picture, because it is now also the First World, since we are systematically transforming the First World into another Third World by our own electronic technologies. If we keep that in mind we can then turn to the Third World for enlightenment, just as Parry and Lord turn to Yugoslavia for enlightenment on the Homeric Epic. They discovered that they performed exactly the same way our jazz musicians perform. (UNESCO statement, 1976)