That date relies on a 1952 reference, and radiocarbon dating was just barely invented, so I would not trust that date estimate of 1000 B.C.
Closest I have heard of is 4,000 years ago, when pygmy mammoths still existed on a small island in the Arctic Ocean. They were quickly wiped out when people arrived.
I don't think the US had many that survived much past 10,000 years ago. When the glaciers retreated and people spread, most of the megafauna didn't last long.