In late September and October, the moon's path in the eastern sky makes a shallow angle with the horizon around the times of full moon. This can cut the time between moonrises on successive evenings to 20 minutes or so on several evenings after full moon. Because of the moon's shallow angle with the horizon, the Earth doesn't have to turn as far eastward each evening to bring the moon into view on autumn evenings. Therefore, evenings around harvest moon time are flooded with more continuous moonlight over several evenings. At more northerly latitudes this time is even less.
Blue moon refers to a second full moon in a given calendar month.
The next time that happens is November when the moon is full on Nov. 1 and again on Nov. 30. Each full moon in a given month has been assigned a traditional name and old almanacs often list some of these distinctive lunar names. Ottewell's list appears below. He offers no guidance in naming a second full moon in a given month, but most months have multiple names for the same full moon so we can take our pick. The upcoming July full moon on July 5 will undergo a partial eclipse but you'll have to be in southeast Asia or Australia to see it.