I've mentioned a couple of times that there was a previous Mudcat discussion of May songs. I haven't found a thread specifically for June songs; a search gave me a long long list of DT songs which mention June. Even "April come she will"(in June she'll change her tune) is included.

But there are a handful of Song a day threads and I noticed, but have not yet looked at, a recent - or recently refreshed? - months of the year inclusive thread.
Jon Boden: 'A Folk Song A Day' blog-finished!!!! includes these lists, previously posted by Joe Offer in Jan 2011. There are blue clickable links on the original thread. But not all the songs selected were directly connected with the months, I think, but many are.:

the May songs:
May 1: Hail, Hail The First Of May (Dave Webber)
May 2: Hal And Tow
May 3: The Outlandish Knight
May 4: The Verdant Braes Of Skreen
May 5: Bonny Bunch Of Roses
May 6: Broomfield Hill
May 7: Death and the Lady
May 8: Gathering Rushes (Underneath Her Apron)
May 9: [Heather] Down the Moor
May 10: As I Roved Out
May 11: Searching For Lambs
May 12: My Husband’s Got No Courage In Him
May 13: Sweet Nightingale
May 14: The Gardener
May 15: Oh What a Beautiful Morning
May 16: Charming Molly
May 17: P Stands For Paddy I Suppose
May 18: Gypsy Rover
May 19: Lillibulero
May 20: Big Rock Candy Mountain
May 21: Roseville Fair (Bill Staines)
May 22: The House Carpenter
May 23: Farewell to the Gold (Paul Metsers)
May 24: Little Sally Racket (Haul 'er Away)
May 25: The Rout of the Blues
May 26: The Banks of the Ohio
May 27: The Birds Upon the Tree (not at Mudcat)
May 28: Babylon / The Bonnie Banks Of Fordie
May 29: Icarus (Anne Lister)
May 30: Little Musgrave And Lady Barnard (Child #81)
May 31: Claudy Banks

Here are the June songs:
June 1: Just As The Tide Was Flowing
June 2: The Beggar Laddie (Child #280)
June 3: Garden Hymn (not at Mudcat)
June 4: Streams Of Lovely Nancy
June 5: Let Union Be
June 6: After The Ball
June 7: Innocent When You Dream (Tom Waits)-not at Mudcat
June 8: Poverty Knock
June 9: Abe Carman from Bellamy's The Transports
June 10: Lock Keeper
June 11: The Ghost
June 12: The Exile Song
June 13: Wild Mountain Thyme
June 14: Rag Fair
June 15: Rose of Allandale
June 16: Farmer's Boy
June 17: Grand Conversation With Napoleon
June 18: The Eighteenth Day Of June
June 19: Roll Alabama Roll
June 20: Bold Archer
June 21: The Way Through The Woods (Kipling/Bellamy)
June 22: Banks Of Sweet Primroses
June 23: When Fortune Turns the Wheel
June 24: The Larks They Sang Melodious
June 25: Farmer’s Toast
June 26: Young Roger Esquire
June 27: One Night As I Lay
June 28: Earl Richard
June 29: White Cockade
June 30: Rose In June