Spain was mainly interested in the southeast and especially Florida because the presence of other Europeans there would threaten the sea lanes between Mexico and Spain. Spain made several attempts at colonizing Florida, including the de Luna expedition, but only got truly serious after the French Huguenots tried to set up a colony in the area.
If there had been continuing French (or British) interest in the area in the late 1560s to early 1580s, Spain might have sustained the resource investment that led to them colonizing Florida and sporadically pushing north as far as South Carolina and into the interior of the southeast. By the time Jamestown was founded, Spain was in sharp enough decline that they weren't able to mount much of a response.
Consequences of a more aggressive Spanish push into the southeast: The Indians of the interior southeast would have probably suffered much the same fate as the Indians of Florida did. They would have been connected more directly to the pool of European diseases that historically swept in waves from Cuba up through Florida and virtually wiped out the Florida tribes. Instead of consolidating into the five civilized tribes, the Indians would probably have been nearly wiped out anywhere that Malaria could spread, and greatly reduced even in the drier mountainous areas.
I've never seen an analysis of how far into the interior southeast epidemics from Florida reached historically, but based on the pattern elsewhere there were probably sporadic epidemics that reduced populations, but spared some areas and were spaced widely enough that the tribes affected had time to recover to some extent.
Archeological studies of the protohistoric period in the interior southeast (roughly the period between Desoto and the advent of British and French traders in the late 1600s) show 150 years of gradual simplification of the Indian cultures. The Indians lost craft specializations when the specialists died out or the population got low enough that it could no longer support the specialists.
The historic southeastern Indians were essentially Mississippians with the top part of their social hierarchy and culture sawed off. The relationship was somewhat like the US as it is versus the US with 90% of the people in Washington to Boston corridor gone, 90% of the people in the thickly populated areas of southern California and Chicago plus burbs gone, and maybe half the population of the rest of the country gone. They were the same people, but without their political/intellectual elite.
Put the Spanish further into the southeast and the Indians there would probably have been reduced to the point where tribes from the periphery would have flooded in.
What would that have meant for British and French colonization? I doubt that South Carolina would have been established in its historic form. Virginia might not have either because Spain might have still been locally powerful enough to strangle it at birth. Beyond that, I don't know. The vacuum in the southeast would undoubtedly have been filled somehow. Maybe Iroquois-speaking tribes like the Erie would have pushed in, or maybe the Shawnees and their relatives. I could even see tribes from the plains filtering into the vacuum. Apaches and Comanches raiding into Georgia and Virginia? Probably not, but interesting to toy with.