Although the title is about clothing, the song is about taking care of yourself, and buttoning up you overcoat against getting cold. Originally written and published in 1928, it was used in the Broadway musical Follow Thru in 1929 starring Jack Haley and Zelda O’Neal who reprised the song for the 1930 technicolour film version, released September 27, 1930, seen here:

Listen, big boy
Now that you got me made
Goodness, but I’m afraid
Somethin’s gonna happen to you

Listen, big boy
You gotta be hooked, and how
I would die if I should lose you now

Button up your overcoat
When the wind is free
Take good care of yourself
You belong to me

Eat an apple every day
Get to bed by three
Oh, take good care of yourself
You belong to me

Be careful crossing streets, ooh, ooh
Cut out sweets, ooh, ooh
Lay off meat, ooh, ooh
You’ll get a pain and ruin your tum-tum

Wear your flannel underwear
When you climb a tree
Oh, take good care of yourself
You belong to me

Button up your overcoat
When the wind is free
Oh, take good care of yourself
You belong to me
Boop-boop-a-doop

When you sass a traffic cop
Use diplomacy
Just take good care of yourself
You belong to me…