Please pardon me if you already know most of the following info. I'm giving it because some readers here probably don't already know all of it.
That specific form of button dates back to the 1700s ...but it is also still in use today, mainly on Military School cadet uniforms (such as the Virginia Military Institue cadets). That being said, the form of button you found was most-widely-used in the first half of the 1800s, and rapidly fell out of popularity afterwards, due to the advent of low-priced "more ornate" buttons.
Back in that earlier era, the specific form of button you found was called a ball-button or a bullet-button. It was frequently worn on civilian clothing, not just on military uniforms.
We diggers date that form of button by various small differences in the way it is constructed, such as its attachment-loop, and the Maker's-Mark (on buttons, it is called a backmark).
The ball-button is "probably" associated with the time-period of the other relics you found at that site. Does your ball-button have a backmark? If so, what does it say?
Edit: I see that Nova Treasure has already given you some of the same information while I was busy typing this reply.