They did, in fact, send expeditions northward. A fellow named Juan Pedro established a handful of settlements in Tennessee and the Carolinas, but these were eventually reclaimed by local inhabitants. The forts as far north as North Carolina were at one time of uncertain veracity, but recent excavations have uncovered solid proof of their existence. By the time they would have rebuilt them, they would have to compete with both English and Americans for the land.

You'll probably be interested in this post that I made a couple months back. It mentions several of the Spanish's failed attempts to colonize the American Southeast. At the time, I neglected to mention two other major failed attempts conducted by Tristán de Luna in the late 1550s and Juan Pardo in the late 1560s. Now's as good a time as any to correct that omission.

Tristán de Luna was a veteran of Coronado's entrada through the Southwest. He teamed up with some of the survivors of de Soto's entrada in an effort to establish a colony at Ochuse (Pensacola, FL). He set sail from Mexico with 13 ships and 1500 colonists and soldiers. After arriving at Ochuse, he sent one ship back to Mexico to bring word of their safe arrival. He waited to unload the remaining ships until his scouts returned with report of the surrounding area, so they'd know the best place to establish their colony. In the intervening time, a hurricane struck and destroyed his ships and their cargo. The scouts returned having found only one village in the area which was unwilling to help support the Spanish. De Soto's men remembered Coosa to the north, which had largely been friendly to the Spanish before, and de Luna ordered a force to march north to reestablish contact with Coosa. When they reached Coosa, at least one of the men thought they were in the wrong place. The region had been devastated by the avarice of a "certain captain," presumably de Soto. The Spanish stayed in Coosa for a time and allied with Coosa to subdue Napochie, a tributary that had become rebellious as Coosa's power declined. The Spanish claimed a large supply of Napochie's maize as their reward and returned to the de Luna's colony. With this and additional supplies that returned with the ship de Luna had sent back to Mexico, the colony managed to survive its first winter, but internal disputes and the continued inability to secure more reliable supplies meant most of the colonists, de Luna included, abandoned the colony when Ángel de Villafañe (a conquistador who spent most of his time salvaging shipwrecks and rescuing castaways) offered to take them elsewhere. Only 50 men remained, and even they only lasted a few more months until they received orders from the viceroy of Mexico to abandon the colony.

/u/CommodoreCoCo already mentioned Juan Pardo's expedition briefly. He established the colony of Santa Elena on the southern coast of South Carolina and was charged with the creation of an overland route to Mexico (of course, the Spanish didn't fully understand the geographic situation and Pardo's task wasn't feasible to begin with). From Santa Elena, Juan Pardo and his men pushed northwest into the interior, establishing several forts in South Carolina, North Carolina, and Tennessee. Along the way they began to here rumors that Coosa was amassing a sizable for to oppose any further intrusion. Before Pardo could have a confrontation with with Coosa, he received news of a French assault on Santa Elena. He left 120 men to guard the forts and raced back to defend Santa Elena. The French attack never materialized, but while Pardo was away, the interior forts were assaulted by indigenous forces (presumably the Coosa and its allies). Of the 120 left behind, there was one known survivor who made it back to Santa Elena to report on the devastation. Santa Elena itself fell to the Cusabo in 1567, though the Spanish re-established the town a year later. After additional attacks and waning interest in the area, Santa Elena was completely abandoned in 1587.