Your US Post Office Department button is shown in the Albert book on American Uniform button in the "Federal Departments" section as button FD-67A. The emblem on it, showing a walking postman, was in service from the early-1900s to 1971, when the present-day "US Mail" emblem was adopted. (The 1800s version showed a postman riding a running horse.)
If your button has a backmark, please post a well-focused closeup photo of it, because that could help us narrow down your button's date-range.