All this talk about where the Crusaders will find food gives me an idea. I hope the Mississippians were around then.
Richard Lion-Heart sets sail for Outremer. They get lost, then get caught in a storm. Richard, along with several other ships, makes landfall in the New World, but not too close to any major population centers. His first look-out is to the health of his troops. They make a permanent camp in a defensible location, and organize scrounging parties. They are delighted to find abundant game. Fear of the unknown keeps desertions down to being with. The natives are somewhat frightened of their strange new neighbors and hence do not force a confrontation. Some trade is conducted during the first winter and the Crusaders learn how to communicate with the natives. Stories are exchanged and the Crusaders learn of a rich, powerful people far to the west. Prestor John, perhaps? Discipline begins to break down in the later winter as hunger comes on and a rogue leader raids and wipes out a nearby town. As news of the massacre spreads among the natives, the area becomes a bit too hot for comfortable habitation, so the Crusaders set off to the West to find the rich cities they had heard about. They shelter in the Appalachians for the remainder of the winter, and set off again after the spring rains end. When they find the cities of legend, they're disappointed to find a bunch of heathens instead of a lost Christian king. What little relations there are go south fast, and the natives ambush the Crusaders one night. This is rather the last straw for good King Richard. The Crusaders open that good ol' can o' conquistadore whoop-butt, and take over the city. They intermarry with the natives, teach them some metal working, how to make cross-bows, etc. Their city soon defeats others, and the new empire stretches far along the Mississippi.
Oops, i forgot horses. Most of the horses are consumed on the voyage, but Richard decides this will not happen to his horses. Guard is mounted, and a dozen or so horses make it to America. They are fruitful and multiply, but become seen as a royal animal. Escapees and the decline of the empire eventually erode that more.