"Meet Me by the Moonlight Alone" is based on J. Augustine Wade song circa 1812. The Carters recorded their version in May 1928, almost 3 years after the similar huge hit song, "The Prisoner's Song" was recorded by Vernon Dalhart. Certainly both songs evolved through the folk process to become distinct songs 100 years later.

MEET ME BY MOONLIGHT (Mr. J. Augustine Wade)

Meet me by moonlight alone
And then I will tell you a tale.
Must be told by the moonlight alone
In the grove at the end of the vale.

You must promise to come, for I said
I would show the night flowers their Queen
Nay turn not away that sweet head-
'Tis the loveliest ever was seen.

Oh! meet me by moonlight alone
Meet me by moonlight alone.

Daylight may do for the gay-
The thoughtless, the heartless, the free-
But there's something about the moon's ray
That is sweeter to you and to me-

Oh! remember, be sure to be there,
For tho' dearly a moonlight I prize,
I care not for all in the air,
If I want the sweet light of your eyes.

So meet me by moonlight alone,
Met me by moonlight alone.

Here's the Carter's version of The Prisoner's Song which could be based on earlier versions by Whiter 1925 or Earl Johnson (In The Shadow of the Pines):

MEET ME BY THE MOONLIGHT- Carter Family Songs

1. I'm going to the new jail tomorrow
To leave the one that I love
To leave my friends and relations
And, oh, how lonely, my love.

CHORUS: Meet me by the moonlight, love, meet me
Meet me by the moonlight alone
For I have a sad story to tell you
To be told by the moonlight alone.

2. My parents, how cruel they treat me
They drive me away from their door
If I live 100 years longer
I'll never go back any more.

Chorus

3. If I had a ship on the ocean
All laden and lined with pure gold
Before my darling should suffer
I'd have that ship anchored and sold.

Chorus

4. If I had the wings of an angel
O'er land and sea I'd fly
I'd fly to the arms of my darling
Then I'd be willing to die

Chorus

There's a thread in the DT with other examples including a version collected in 1924 by Vance Randolph. You can also find posts on The Prisoner's song and discussion. No sense copying too much info here.