I can't help but wonder if "Walk Right In..." is derived from this song:

From the sheet music at Brown University:

"Bert A. Williams' Big Hit"
COME RIGHT IN, SIT RIGHT DOWN, MAKE YOURSELF AT HOME
Words by Bob White & Alfred Anderson, music by Will H Dixon, ©1909.

1. Rufus Johnsing Snowball Lee was known both near and far
To be the greatest ladies' man upon a private car.
He had a gal in ev'ry town who met him at the train,
And when the boss was out of sight, to her he would exclaim:

CHORUS: Come right in, sit right down, and make yourself at home.
You've found the place you're looking for; there's no more need to roam.
I know you're hungry; eat your fill,
For Mister Pullman pays the bill,
So come right in, sit right down, and make yourself at home.

2. Rufus quit the Pullman job to go out with a show.
They only played a night or two, then busted up, you know.
While walking down a railroad track he spied a house nearby,
And when he knocked and asked for grub, the keeper made reply:

CHORUS: Come right in, sit right down, and make yourself at home.
You've found the place you're looking for; there's no more need to roam.
The sign reads "Small Pox" on the door,
But a welcome mat lies on the floor,
So come right in, sit right down, and make yourself at home.

3. Rufus spied a stack of hay and fell in for a snooze.
A stray spark fell in also and the flames began to ooze,
Which started him to dreaming of a region far away,
And as he snored, he thought he heard the gay old Satan say:

CHORUS: Come right in, sit right down, and make yourself at home.
You've found the place you're looking for; there's no more need to roam.
The world's so awful cold, I know,
But down here there's no fear of snow,
So come right in, sit right down, and make yourself at home.