Desperate for escape, some Huguenots sought refuge across the Atlantic Ocean.
This needs a lot of explaining! In fact I suspect an independent French Huguenot colony without the patronage of some other European power nor the protection of the French king is categorically impossible. It is quite possible and OTL for the Huguenots to seek the protection of Protestant powers and even to then be routed to a colony somewhere overseas.
They did of course try exactly this OTL, in both Brazil and Florida. In fact it seems that the reason the Spanish outpost of St Augustine came into existence was precisely to expell such an attempt, at a site today unknown precisely but somewhere near the modern OTL Jacksonville. But the "colony" numbered in the low hundreds and was on the edge of extinction even before the Spanish expedition sent to remove them largely massacred them.
Now I can imagine that if politically expedient (that is, the King of France was at war with Spain or didn't mind risking it for a purpose) that the King might decide to avenge the Spanish massacre of his Protestant subjects, and double down (rather, raise the stakes by orders of magnitude) with a project to first turn the tables on the Spanish by investing and expelling the Spanish fort a-building at St Augustine, take it over and pour in a major expeditionary-colony, and invite Huguenots to settle there. He would be foolish to trust entirely to their loyalty of course and I think a possible model for Huguenot Florida could be that while the majority of settlers are Huguenots who enjoy settlements entirely of their own numbers, they agree to support other settlements nearby of predominantly Catholic subjects who are dedicated to supporting Catholic army and navy forces to supplement the Huguenot militia, the two branches of force being unified in command and accustomed to joint operation. Thus it would be a mixed Huguenot/Catholic colony, though the King would be careful to pick Catholic leadership pledged to tolerate and work with the Huguenots without invidious prejudice.
The purpose of state of the King doing this would be two-fold; on one hand, it is a safety valve, to defuse surges of anti-Protestant terrorism (and contervailing Huguenot counterattacks which were quite as terrible) by pressuring Huguenots to emigrate; on the other hand, if he can manage to keep this colony loyal, he can use it to threaten and distract the Spanish giving cover to the Catholic-only New France colony far far away to the north. (This might help explain why the Huguenots don't expand as rapidly northward as one might expect, they are encouraged to direct their expansionism southward to try to get footholds on the Gulf of Mexico and ideally to take control of the whole Florida peninsula so as to secure passage of French ships into the Gulf.
I amused myself by imagining that the French having taken St Augustine (with substantial part of the force doing it Huguenot volunteers, though not the whole of it!) the Protestants choose to maintain and expand the Spanish site rather than return to their own razed one, and even keep the name, in French of course (Saint-Augustin) since St Augustine of Hippo is I gather a much revered authority in Calvinist doctrine; retaining the name chosen by the murderous Papists is a sobering memento mori to inspire the Huguenot settlers to answer the call to royal service in defense of their new homeland.
Looking at it long term:
1) it is pretty Utopian to suppose that the OTL decimation and exile of the Huguenots from France can be sidestepped by "peacefully" transferring them wholesale to Florida. Over time the rate of emigration across the Atlantic would increase of course, but we are talking about the mid to late 16th century here, and then the 17th century during which time the position of the Huguenots in France was worsening versus their high water mark of the Edict of Nantes. Early on, it is impossible to ship 2 million persons from France and dump them on the wild shores of North America anywhere with any expectation they survive. The shipping doesn't exist to carry them in the first place anyway. Any Floride colony must be built up gradually with more judicious injections of immigrants at a sustainable pace.
2) the general class structure of the Huguenot movement in France doesn't support them all freely agreeing to be settler colonists. Apparently conversion to this French form of Calvinism was a matter of nobles, gentry and persons higher up the social ladder generally. Few to none of them were peasants! Whereas a settler colony needs people who are going to get out there and till the soil, as well as stand by ready to take up arms against Native foes resentful of their seizure of land backed up by Spanish power. The Huguenots OTL had no problem creating quite effective armies, but pioneers they generally were not, by inclination. In fact a considerable number of them did wind up emigrating to varous pioneer situations, but these I believe were a minority and also driven by desperation. Whereas if the mass of Huguenots in France are driven by desperation indeed, by being crushed in civil war, they hardly have the opportunity to muster an orderly colonization effort!
I can foresee the possibility of the French Florida colony as a reserve outlet for Huguenots being kept under French Royal protection for a century or so without deviation from OTL, bringing us to the late 17th century, by which time the colony can be very populous and a major military force that Spain cannot hope to destroy. But under these circumstances, the Huguenots would be more powerful back in France than OTL--perhaps not so much so that Louis XIV would not change his mind and attempt to repress them as OTL.
If the Bourbons follow their OTL anti-Protestant course, then obviously there must be a breach in relations between La Floride and France--either the French come in with a really huge army, and massacre/scatter the colony as well as of course the remaining Huguenot population in France--which, news flash, is not likely to be a lot smaller than it was at this point OTL!
Or--considering the utility of the Florida colony, and that if a compromise going on a century or so can merely be continued, they continue to be of use to his policy, the King may instead resist taking a hard line against these Protestant subjects and keep stringing them along. Perhaps a combination of incremental discouragment in France and open opportunity in (southeastern) America (not in New France and perhaps not in Louisiana or Haiti either) has indeed caused the remaining Huguenot population in France to dwindle. Even so, it will not do for the King to take too hard a line even just rhetorically.
More likely, if a royally supported and protected Florida colony can be sustained, the Huguenots cannot be too disabled in France itself and therefore will not dwindle.
Assuming the French kings must bite the bullet sooner or later, "cleaning up" Florida is going to be a headache for whichever one pays the butcher's bill, and part of his reward will be to scatter greater numbers than OTL of sworn foes to shelter in the protection of all France's enemies.