"that he may" or "that he can" with the same meaning as "so that he may/can" seems significantly old-fashioned to me also, but Google ngrams shows it still in use, albeint declining, through 2000. A few examples:
- Thus, not only that man may know the purpose of his life, but also that he may know the nature of the life through which he must... (A Map of life F. J. Sheed - 1994)
- that he may find his own reasons to doubt Christian teachings (This Month magazine)
- A patient is said to submit to cautery or lancet that he may obtain health. (Philosophic Classics)
Other examples were largely in quotes or translations from ancient authors, or in religious works where no doubt the language of the KJV (King James Version) was influential.
I have mostly seen this used (I don't think I have ever heard it in informal conversation) where there seemed to be an effort to assume an antique air, or a more or less intentional imitation of biblical language, particularly of the KJV.