I don’t remember reading of any horses getting loose till after the Puebelo revolt in 1680, Santa Fe. Hogs yes! Horses, no.
Ever try to keep track of your horses before you have fences?
Trust me there were already wild horses in North America within a decade of the first landings on the continent (in Mexico), and certainly there were horses in the offshore islands first settled by European.
We have an ancient group of ponies at Assateague and Chincoteague islands in Virginia ~ which are easily traced to 1600 and probably earlier.
The pirates who sailed the Spanish Maine were headquartered in Chesapeake Bay in the 1500s. There were Spanish settlements here. By 1598 the Spanish made it to Santa Fe New Mexico on horses that'd been born in America of lines that'd been here upwards of 75 years.
Never fear, Indians knew how to steal and the Comanche had Spanish horses available to them when they still lived in Montana!
De Soto brought HUNDREDS of horses with him for his expedition to the limits if LA Florida ~ including to the great inland sea (Lake Michigan). Some of them got away. Actually almost all of them got away!