Mods, this is my first attempt at an answer here - please let me know if this isn't up to standard.

While I can't speak to other southern/southwestern states, for Louisiana the answer is that it went through a unique entry into the United States that specifically preserved the civil laws in force under its prior ownership.

The reason Louisiana has civil law, stretches back to 1712. That year, the French king issued the first set of "Letters Patent" which made the laws in force in France at the time applicable to the then-colony. That included the French civil law.

But in 1762 France ceded the colony to Spain, with the first Spanish governor taking power in 1766. That didn't immediately change anything due to resistance from the French population, but in 1768, after putting down a rebellion, the Spanish General Alexander O'Reilly began a reform of the existing systems of the colony under a mandate from the Spanish crown. It appears from the available documents that O'Reilly had put Spanish law into effect in the colony by 1769.

In 1800, Spain and France signed a treaty in which Spain ceded the colony back to France. But the countries agreed that the transfer would remain secret as long as Britain and France remained at war, so the official transfer did not go forward until 1803. The general agreement is that the laws in force were never transitioned back to the prior French law, because France had assumed sovereignty for only about 20 days before it sold the territory to the United States through the Louisiana Purchase on April 30, 1803.

The United States assumed control of Louisiana in December 1803, and one of the first official acts of the new commissioner William C.C. Claiborne was to provide for the retention of the "laws heretofore in force" in the territory - the Spanish laws. As a lawyer from a common law tradition Claiborne was strongly opposed to continuing the civil law system in Louisiana, but in 1806, under continuing political pressure on the subject, by-then-Governor Claiborne eventually approved a resolution by the new Louisiana legislature appointing two lawyers "to compile and prepare jointly a Civil Code for the use of this territory." Not only did the resolution require development of a civil code, it directed that the appointed lawyers "shall make the civil law by which this territory is now governed, the ground work of said code." Two years later, on March 31, 1808, the legislature formally adopted the new civil code, and although there have been a few substantial revisions of the code since, Louisiana has never switched to a common law system.

So the short answer is that when Louisiana was purchased by the US, the laws that were then in force - civil law, and specifically a Spanish version of civil law - were initially retained for the sake of stability, and then reworked and officially adopted in the territory due to popular political support.

As a side note, one of my law professors - a very stern Frenchman - was unusually emphatic that Louisiana law is based on the Spanish civil code. Why would a French guy be so adamant that Louisiana's civil code didn't derive primarily from French law? Because one of the lawyers appointed to develop the new civil code, Louis Casimir Elisabeth Moreau Lislet, was also a Frenchman, and no honorable Frenchman would have disobeyed his directive to model the new code on the laws in force at the time - the Spanish civil code.

Sources: The History and Development of the Louisiana Civil Code, John T. Hood, Jr., Louisiana Law Review, 19 La. L. Rev. (1958).

An Elementary Treatise of the Civil Law of Louisiana, John Randall Trahan.

Further reading: Moreau Lislet: The Man Behind the Digest of 1808, Alain Levasseur, Claitor's Publishing Division, 2008.