This seems to be a bit older than Jimmy Rogers' recording.

Wehman Bros.' Good Old-Time Songs No. 2 (New York 1910) has on p.70

Hush-a-Bye, Baby

Now show me the lady that never would roam,
But would stay with her family at night,
And never go roaming after the boys,
But would sit by her fireside so bright.
My wife she is one of those different kind,
And often she causes me to weep,
By the way she does roam and leaves me alone
To rock the dear baby to sleep.

Chorus:
La la, la la, hush-a-bye, baby,
Chucking the baby ever so high;
La la, la la, hush-a-bye, baby,
Mamma will come home to you by and by.

We've a neat little cottage all shaded by trees,
As happy as happy can be,
Where we laugh and we walk, over baby we talk,
For none are so merry as we.
He's just six months old, and he tries hard to scold
By clapping his fat, chubby hand,
And the baby does cry when any one's nigh,
He's the sweetest dear babe in the land.