My mother sang a version that had a introduction with a different tune. I only remember fragments:

Here comes the sandman
Tripping so lightly
. . .
And he scatters the sand
With his little brown hand
In the eyes of the sleeping children.

Go to sleep, my baby.
Close your tired eyes.
The . . .
. . . paradise.
The little stars are peeping
To see if you are sleeping.
Go to sleep, my baby,
. . . .