I'd tend to say "yes, the Great Vowel Shift operated on non-primary-stressed syllables," but there are several factors that make this question difficult to answer. I don't think that "hypercorrection" is really a useful concept in discussing the history of /iː/, /əɪ/, /aɪ/ in non-primary-stressed syllables.
Middle English words may have been stressed differently, or even divided differently (as pointed out by tchrist in a comment.)
Middle English /iː/ was mainly found in primary stressed syllables. One reason for this is that, according to Alex B.'s answer to the question "What did we gain in return for the loss of phonemic vowel length from Old English?",
long vowels were always stressed in OE - in unstressed syllables, long vowels got reduced.
Because of this, in words that have native English etymologies, I was only able to find non-primary-stressed /aɪ/ in compound words. There is evidence that suggests that some of these were already treated as single phonetic words in Middle English; words such as twilight and insight can be found written together in many ME texts. However, it is quite possible that the compound nature of these words caused them to forego normal development according to sound laws; they may have acquired diphthongs by analogy with forms such as light and sight.
Reduced forms are also attested for some elements of compound words, but it's not clear to me if reduction was common enough to call it the phonetically "regular" result. Reduction formerly occurred in many words ending in -wife such as midwife (the pronunciation /ˈmɪdɪf/ is recorded, although it's unclear if it was ever the main pronunciation), hussy (from housewife, which has had the variant pronunciations /ˈhʌzwɪf/ and /ˈhʌzɪf/ ) and goody (from goodwife). This is perhaps comparable to the weakening of the vowel in many words ending in -man.
(We can also compare the modern decline of these reduced forms to the decline in some other reduced pronunciations such as "forrid" for forehead, "wesket" for waistcoat and the like.)
Finally, there are learned words taken from French or Latin; Peter Shor listed a few in a comment (and also wrote a relevant answer to the linked question):
triumphant, glorify, terrify, porcupine, among others. There are
lots of examples of words which had /i/ before the Great Vowel Shift
and currently have /ai/ in an syllable without primary stress.
One complicating factor here is the position of primary stress; I suspect some of these words may have had stress on the last syllable in Middle English (Middle English stress is discussed some here: Middle English Phonology).
Another important complicating factor here is that learned words are constantly being "re-learned" to some extent; you could call this hyper-correction, but it's even more pervasive than that term would imply. For example, a word such as arctic, which lost the c in Middle French, can have it re-introduced into English spelling and subsequently pronunciation. The re-introduced /l/ in words like false, fault, vault is now nearly universal, even among speakers who have "fawcon" for falcon. I wouldn't say these are "hyper-corrections" in Modern English, and I think it might not even be quite right to say that they originated as hyper-corrections. It's true the modern pronunciation of words like these has been influenced by the spelling, but this may have been the case for as long as the word has been in English. My point is that words like this are not normally transmitted solely by mouth and ear. In situations like this, it seems to me that we can't fully describe the way pronunciation evolves without considering written as well as spoken forms.