If you want something you can be reasonably sure nobody else is singing, look at "Pink Taffeta Sample Size 10", a Cy Coleman/Dorothy Fields number that was cut from SWEET CHARITY. A number of ladies have recorded it over the past 50 years, including Mimi Hines and Debbie Shapiro Gravitte.
You'll have to cut it, I'm sure. It's also a period number, since we don't have traveling ladies' dress salesmen (like the singer's father) any more; but the context is explained in the lyric and it doesn't take a genius to figure it out. I assume it was originally written for Gwen Verdon, so don't expect any vocal pyrotechnics (unless you add them yourself). But it's a sweet song, lots of character details, and not often performed.