Sailing across Atlantic.
Quite likely different people used different routes, just as the Europeans did after Colombus. Also there is some shifting of the currents as the earth gets warmer or cooler. Ancient man was probably much better at living from materials harvested from the sea. The Spaniards were already too "civilized" to be so good at that. The great polynesian voyages were much longer than Atlantic ones and they managed to transport women children and livestock and settly many of the Pacific islands.
In terms of our own history. The Spaniards settled the Caribbean and the south using the Equatorial Current. The English took the northern current and ended up in New England instead of Virginia where they were trying to go.
But we're discussing crossing the Atlantic from Spain to the New World via the Equatorial current. It took Columbus 33 days at an average of ~5.4 kts before landfall. That's 4,200 nautical miles.
Even with a generous 1kt equatorial current, you're talking nearly 6 months at sea. And contrary to your belief, there's very little to 'harvest' once you get beyond the continental shelf.
Certainly Thor Heyerdahl crossed the Atlantic from Morocco to Barbados (3,300 miles. 57 days) in a reed boat. But they used sails and were provisioned with food and water. A few things I doubt pre-agricultural humans had.