Ordinarily when a population outgrows its multi-generational territory, it either makes a deal with the neighbors, or gets into a conflict with them. Precolumbian America was no different than any other place on Earth. Given the number of language families, I very much doubt that the Americas were settled in one go, by just one pretty small cultural group.
OTOH, I’d think the Polynesian expansion was probably merely the last coat of paint. At least some of the places they took over had been occupied before, but being islands, if the prior population had managed to lose inter-island mobility, it was bound to be a fight.
https://en.wikipedia.org/wiki/Classification_of_the_Indigenous_languages_of_the_Americas
I'm in total agreement there, particularly considering the persistence and spatial distribution of so many distinct language phyla. I suspect the maritime migration was one of the first. You'll recall that quote I posted from Juan Crespi about the song sung by maritime people of Queen Charlotte Is. as being the same as sung all the way to San Diego despite the distribution of those remote language families: