It seems to be an idiosyncrasy of Poe to use a definite article when there's an adjective preceding the month:
The Island of the Fay:
I came upon them suddenly in the leafy June, and threw myself upon the turf, beneath the branches of an unknown odorous shrub, that I might doze as I contemplated the scene.
To -- -- --. Ulalume: A Ballad
The skies they were ashen and sober;
The leaves they were crispéd and sere—
The leaves they were withering and sere;
It was night in the lonesome October
Of my most immemorial year;
In The Bridge of Sighs, there's no article, presumably because there's no adjective.
The bleak wind of March
Made her tremble and shiver;
I don't think there's any explanation beyond that, given that I've not seen this used elsewhere. (Edit: Now I have seen some examples, and I can't think of an explanation for why.)