The authorship of the "little tip" verse is nicely sidestepped in the Smithsonian/Folkways Ella Jenkins Camp Songs compilation album:

” It is one of Ella’s favorite songs, and she sings it with Bucky Halker on an album built around ideas of unity, cooperation, and labor unions called A Union of Friends Pulling Together (SFW 45046). The last
verse was written later, and reflects changes in women’s roles in
the workplace that have occurred since Woody wrote the song

https://folkways-media.si.edu/liner_notes/smithsonian_folkways/SFW45082.pdf