The latest pre-Columbian migration from Asia is relatively recent. We're talking 1300 years ago, or so. Here is the abstract from a study from 2013. The main theme is the evidence for the use of archery in warfare, but it contains other interesting details.
There were at least four waves of bow and arrow use in northern North America. These occurred at 12000, 4500, 2400, and after about 1300 years ago. But to understand the role of the bow and arrow in the north, one must begin in the eighteenth century, when the Russians first arrived in the Aleutian Islands. At that time, the Aleut were using both the atlatl and dart and the bow and arrow1 (Fig. 1). This is significant for two particular and important reasons. First, there are few historic cases in which both technologies were used concurrently; second, the bow and arrow in the Aleutian Islands were used almost exclusively in warfare. The atlatl was a critical technology because the bow and arrow are useless for hunting sea mammals. One cannot launch an arrow from a kayak because it is too unstable and requires that both hands remain on a paddle. To use an atlatl, it is necessary only to stabilize the kayak with a paddle on one side and launch the atlatl dart with the opposite hand. The Aleut on the Alaska Peninsula did indeed use the bow and arrow to hunt caribou there. However, in the 1,400 km of the Aleutian Islands, there are no terrestrial mammals except humans and the bow was reserved almost exclusively for conflicts among them. The most significant event in the history of the bow and arrow is not its early introduction, but rather the Asian War Complex 1300 years ago, when the recurve and backed bows first entered the region, altering regional and hemispheric political dynamics forever.
Source: The Bow and Arrow in Northern North America
HERBERT MASCHNER AND OWEN K. MASON