There's another one from 5 years later in 1913: https://lyricsplayground.com/alpha/songs/a/alwaystakeagirlnameddaisy.html

ALWAYS TAKE A GIRL NAMED DAISY ('CAUSE DAISIES WON'T TELL) (Music: George W. Meyer / Lyrics: Alfred Bryan & Sam M. Lewis)

Jimmie Hodges - 1913 Albert Campbell & Henry Burr - 1913

Handsome Harry, handsome Harry Thomas
He was sued, yes, sued for breach of promise
He took Mary walking through the dell
And said, "Now don't you dare to tell"

Mary went right home and told her mother
Ma told Pa and Pa then told her brother
Brother told the preacher
And the preacher went and tolled a wedding bell

Never take a walk with Mary
Never take a walk with Sue 
Never take a walk with Maude or Carrie
That's the kind of girl you'll have to marry

If you take a girl out walking
Down a little shady dell
Always take a girl named Daisy 
'Cause daisies won't tell 

Harry's married life was pure and simple
Till he met a girlie with a dimple 
She said, "Dear, I'm not acquainted here 
I just came down from Beaver Fall"

Harry winked and said, "Why silk and satin
To this girl would be like Greek and Latin"
Harry felt like fainting 
When he missed his little dollar Ingersol

Never take a walk with Mary
Never take a walk with Sue 
Never take a walk with Maude or Carrie
That's the kind of girl you'll have to marry

If you take a girl out walking
Down a little shady dell
Always take a girl named Daisy 
'Cause daisies won't tell

That one's clearly not quite as innocent.

So I would guess that it was a phrase from the time when casual sex and premarital sex simply were not socially acceptable, and even being alone with someone of the opposite gender had implications. So people would sneak off and meet up in a field somewhere where they thought no one would see them together and tell.

But that's just a guess based on those two songs.