A Google search of "Songs with June in the title" yields lots - mostly non-folk, but needs must. Some highlights:

http://www.thecavanproject.com/29-songs-about-june/

https://www.quora.com/What-are-some-songs-that-mention-the-month-of-June-in-the-lyrics

https://www.ranker.com/list/the-best-songs-about-junes/reference

(More frivolously, JUNE Tabor has some very nice songs :-))