"The Desrick on Yandro" mentions different songs, despending on the version.
In the 1964 Ballantine Books edition, the songs mentioned are:
  - Rebel Soldier - p. 86
  - Well I Know That Love Is Pretty - p. 86
  - When the Stars Begin to Fall - p. 86
  - "The Yandro Song" - pp. 86, 94

In the original version, the songs listed are:
  - Witch in the Wilderness
  - Rebel Soldier
  - Vandy, Vandy, I've Come to Court You
  - "The Yandro Song"

To be completely frank, I have no idea why they would change something as minor as 3 songs listed before the fifth paragraph of the story from one version to the other--especially since the three changed songs have no relevance on the outcome of the story. For some reason, Rebel Soldier (aka Jack of Diamonds) was the only song mentioned in both versions of the book; it has been discussed earlier in this thread. The Vandy, Vandy song will be discussed later, as it has its own chapter in the book. When the Stars Begin to Fall is a Christian song dating to at latest the 1860s, but published in several Christian songbooks in the 1920s and 1930s. You can read more about it here.

The last two songs listed briefly in the introduction are a bit harder to find. Though Dolly Parton's song "When Love Is New" matches well with the name of Well I Know That Love Is Pretty, I have some doubt that that's the song Wellman was referring to (ok, ok, more than some). I haven't found a song I'm quite happy with to match this song, but I'll assert that it's a song related to O Waly Waly (Water is Wide), an Americanized version known as Young Ladies or Little Sparrows. My only real backing behind that claim (and it's a sort of tenuous one) is that Wellman was aware of a version of the song that went "Oh, love is fair and love is charming, / And love is pretty when it is new" (source). Aside from that, I've got nothing on it.

To be totally honesy, Witch in the Wilderness has got me stumped. Maybe someone else would be able to better identify a song by this name? Though it's tempting to say that it could be the same song as the Yandro song, given the subject of the chapter (a male witch who lives in a shanty in the forest), this is not the case: the song is introduced as follows: "Staying off wornout songs, I smote out what they'd never heard before--Witch in the Wilderness and Rebel Soldier and Vandy, Vandy, I've Come to Court You. When they clapped and hollered for more, I sang the Yandro song, like this...." Again, not an important aspect of the story, but it certainly makes you wonder...

And finally, The Yandro Song, which is known as "He's Gone Away"

"I'll build me a desrick on Yandro's high hill,
Where the wild beasts can't reach me or hear my sad cry,
For he's gone, he's gone away, to stay a little while,
But he'll come back if he comes ten thousand miles. "

"Look away, look away, look away over Yandro,
Where them wild things are flyin'
From bough to bough, and a-mating with their mates,
So why not me with mine?"

The first line of this song matches nearly verbatim a line from the version of He's Gone Away in the DT here at Mudcat, however, aside from a reference in Sandburg's American Songbag, I haven't seen much about this song. In a 1965 article for the Raleigh News and Observer, Wellman hinted that he thought the local residents may have been "putting Sandburg on," so-to-speak, citing Sandburg's 1927 American Songbag publication; despite that, apparently local residents agreed to call a section of Walnut Mountains (Madison Co., NC) "Yandro," though Wellman consented that they had probably meant "yonder" in its original sense of the word.