In Moonlore, by Gwydion O'Hara, I found that the Hunter's Moon follows the Harvest moon because when the harvest was complete people had to turn to the hunt for their main food source. It is also known as the Blood Moon or Killing Deer Moon.
Diana is the ancient Roman moon goddess and also goddess of the hunt. She survives in Shakespereare as Titania the Queen of the Faeries (according to O'Hara) which probably opens up a few song possibilities... But it's late and further research is beyond me..
Tinker