Absolutely! And I know that I've actually answered this question before, but can't seem to find the link.

The easy answer is yes, they definitely did. Alaskan Eskimos in fact remained aware of the other side of the Berring Strait - i.e. it was a body of water that they crossed at times, the same would be true of Inuit living at the Eastern end of the Arctic - Greenland was definitely a known place "across" the Atlantic from them.

I don't know much about Eastern North America, or much about South and Central American Indigenous histories (besides bits and pieces like Viracocha leaving and crossing the Pacific), but I do know that in the Pacific Northwest there are stories of crossing the Pacific, circling the Pacific, or of people coming from across the Pacific.

Part of many origin stories for different lineages begin with a journey to the lands of the copper, abalone, and so on. These aren't always in the same place, sometimes the copper is in the far north, other times it is across the sea, or even to the East. In the Nuxalk story of the Salmon boy there is a good example of at least one of these stories. The Salmon boy and his brother travel to the west, visiting in turn various lands of the fish people which are located out across the open sea, where the salmon people live, who wear salmon skin clothing (McIlwraith "The Bella Coola Indians" and oral stories). This always makes me think of the Ainu, indigenous Japanese who did wear salmon-skin clothing, but the chances are that this story has more to do with the order of fish arriving in the spring.

Then there are other stories that are more historical. There's a story recorded by Cliff Kopas (also a story about the Bella Coola Valley) where it talks of canoes who arrive in the valley, with people in them who wear grass clothing. They stay for several years, then return to where they came from.

I have several personal accounts of Nuxalk people who say that they have met Pacific Islanders who have the opposite story in their lineages, but have yet to find an independent source for it, but between this and the theories of Thor Heyerdahl (who spent time in Bella Coola during the war) regarding connections between the area and New Zealand, the nation here has developed quite a strong relationship with several communities in New Zealand, resulting in a number of marriages and large group trips.

The tribes of northern California had origin myths that made reference to supernatural people/animals/deities from across the sea. These natives had origin myths that were complex and involved many different major characters. Kroeber described origin myths of many of the tribes as being a common spin on the same general theme; a deity or more comes from a certain distant place (often across the ocean), they arrive here and then create the world/places/critters. After creation, the supernatural beings go back home - across the Pacific.

Among the Yurok, one myth tells of the creation of the ocean and acknowledges lands across the ocean. Wohpekumeu, a Yurok deity, was a widower from across the ocean. Among the Hupa a principal deity, Yimantawinyay, is literally "old man across the water he goes". A prominent Hupa Maidu myth recalls when Coyote and another earth maker sing the earth into existence from across the ocean.

See:
Kroeber, A. L. (1976). Handbook of the Indians of California. United Kingdom: Dover Publications.

Buckley, T. (2002). Standing Ground: Yurok Indian Spirituality, 1850–1990. Berkeley; Los Angeles; London: University of California Press. Retrieved April 28, 2020, from www.jstor.org/stable/10.1525/j.ctt1pp695

Edit: Maidu for Hupa

Yes, the Mexica had a myth regarding the Atlantic ( which came from the toltecs, who they saw as predeccesors). For starters, the Mexica called the known world Cem Anahuac which literally means all that is surrounded by water in Nahuatl refering to their lands being surrounded by great seas, also could be interpreted as "reality" or "continent".

While not the main focus, we need to give a short review of a myth: the departure of Quetzatcoatl

In the Toltec civilization one of their myths was that the god Quetzatcoalt came down to Tula, their main city. He taught the people things such as farming, crafts, art, astronomy and is the reason their power and influence grew so much, at the cost of going against the wishes of the other gods who were not as interested in the well being or development of humans

Gods such as Quetzalcoatl's brother, Tezcalipozca, planned to fool Quetzalcoatl into breaking his vows and shame him so he would never return. He disguised himself as an old man and gave him Pulque, a holy alcoholic drink, and in his drunken state Quetzatcoatl had sex with his sister. Ashamed, he started to march into the atlantic sea, and either turning into or making Venus (star of great importance in mesoamerican cultures) and vowing to return someday with his followers.

Now it's to note that variations of this myth are prevalent even among far away maya civilization, where a simimar myth that Kukulkan would return from the sea is present. Venus comes from the East and mesoamerican civilization seems to have valued the eastern ocean as having spiritual value, seen with caribbean starfish making it to central mexico, used for depictions of the underworld.

Toltec civilization had a huge inluence in mesoamerica and the later Mexica culture took these myths and gods as they saw themselves as their succesors and as a form of getting prestige, as such their oral myths were taught at school by elders and everyone no matter social class was required to attend one(tepochcalli or calmecac), as such every Mexica knew of the myth of Quetzalcoatl dissapearing into the endless sea to one day return again.