Thirty days hath September, April, June and November?

According to everything I can find^1,2, and all usages I can remember ever coming across, hath is the 3rd person singular present tense of have, and not the plural.

So why does the rhyme go as follows?

Thirty days hath September, April, June, and November

Should it not be "Thirty days have September, April, June, and November"?

Hath could be singular or plural in Middle English in the southern dialects.

Let's look at the first known English version of the rhyme, and discuss its form according to observed practices at the time. We go back to Harley MS 2341 (fifteenth century), as recorded in Secular lyrics of the XIVth and XVth centuries edited by Russell Hope Robbins (and cross-checked with Wikipedia):

Thirti dayes hath nouembir,

April, iune, and septembir;

Of xxviij(ti) is but oon,

And alle the remenaunt xxx and j.

At this point, November appears first, but even here the lyric preserves hath. The line breaks pose one possibility - to echo other commenters - that November would be read as a separate and more proximal subject to hath, warranting the use of the singular form.

However, this explanation is unnecessarily convoluted. Middle English verbs in plural can take an -eth ending, which in the case of has is hath. Here is Stephen H. A. Shepherd's instructions on Middle English in Middle English Romances (1995; not accessible online) (cross-checked with this course website on Middle English tenses):

The present plural ending eth is southern, whereas the e(n) ending is Midland.

To use Shepherd's examples, he loveth or he taketh would have the same verb ending as we loveth or we taketh in some dialects.

The Middle English Dictionary bears this out for hath, citing hā̆th as both a third person singular and an all-persons plural form of the verb. We are seeing plural hath here.

While in Early Modern English the unmarked plural replaced the -eth form and have replaced hath, the lyric has preserved the older verb usage across the generations.