Thanks to the prompt responses on the clamp or aglet.
I figured out what it was, your help made me look at what I already had, since no one was big on the shoelace idea. What we have from this internment is a pair of suspenders, marked "President"  and "Shirley MAKE."

The aglets (we have two) are from the bottom of a pair of President Suspenders; they drew two cords together right above the tabs for the buttons on the backside. They date from around 1902 (when Sears says they are new for the catalog) and 1917, when the C.A. Edgarton Mfg. Co. company had merged with Boston Belt to form the All Star Athletic Company and was making Chuck Taylors. 

I wonder how long suspenders were kept and worn? They were 43-50 cents for many years. 

An ad showing the suspenders can be viewed at: 
http://www.kaleden.com/search/images.jsp?id=515538

Interestingly, I also found poster that shows a woman baseball player wearing a baseball uniform and a pair of President suspenders. 
Suzanne