Question

Which conquistador was the first to push Spain's explorations into North America?

Answer 1

  You are probably looking for Juan Ponce de Leon 1474 – 1521. He was not actually a conquistador, but a gentleman volunteer on Columbus second voyage, later a governor in Hispaniola and later still an explorer. He explored Florida in 1513. 
Cortes did not reach the mainland until 1519. 
Hernando de Soto came to the new world in 1514. 
Pizarro accompanied Balboa in his crossing of the Isthmus of Panama in 1513. But that is more Central America. 
Cortes, de Soto and Pizarro were actual conquistadors.