There are various archeological sites in North America that have been dated to around 18 thousand years ago (kya) to 12kya, while there are several roughly contemporaneous sites in South America as well. Examples of these sites can be the Folsom and Clovis sites in what is now Texas, and the Monte Verde site in what is Southern Chile. Expansion, especially to such a degree, is usually explained by some geological, ecological, or sociological context. But, considering the relatively low population in North and South America at the time, and the overwhelming prevalence of flora and fauna, even mega-fauna, for hunting, the likelihood that societies could be driven farther south due to resource scarcity seems slim at best. Therefore, what drove such a far expansion to the south? Did the prevalence of resources provide an excess in which exploration could be considered a worthwhile use of the cultures time and effort, or was necessity the driving factor?
;tldr; I can't think of any reasons for such a dramatic expansion south that there seems to have been? Was it due to resource scarcity, or just sheer curiosity?
This might be a better question suited to /r/AskAnthropology
The causal reasons for migration are probably too numerous to list though both of your propositions seem reasonable to me.
For example, Pacific Northwest Coast Indigenous oral histories all speak to a 'great flood'. The 'flood' can be traced in sea level changes during the last Ice Age when the pressure of the ice sheet caused the NA plate to depress below current sea levels (up to 200m in some places see https://www.academia.edu/3913219/Sea_Level_Change_and_Archaeological_Site_Locations_on_the_Dundas_Island_Archipelago_of_North_Coastal_British_Columbia). These 'floods' caused many groups to move significantly inland. In other cases the pressure from the ice sheet caused places, Haida Gwaii for example, to rise up to 200m out of the water forcing residents to move to areas now well below current sea levels. One of the most exapansive archaeological sites in the area (Boardwalk Site in the Prince Rupert Harbour) extends below current sea levels as well.
These same oral histories talk of the effects of invasions and wars that caused populations to relocate (See Susan Marsden's book "Defending the Mouth of the Skeena."), though these events are not as old as the sites you mentioned nor are they on the same scale.
I think the issue is not so much that populations moved from north to south, if I understand your question correctly. There are probably many migrations to North and South America. Some by sea across the Pacific, some following the coast and some more land based. It is also worth pointing out that virtually every Indigenous culture maintains that they have always lived here.