Reconstruction era law enforcement

[Sampe]

this question isn't really related to alternate history but since i spend a lot of time reading these forums, i know there are a lot of knowledgeable people here. I have been thinking about trying my hand at writing a short historical mystery novel set in the early reconstruction era south (mississippi probably), but the problem is i don't know much about how law enforcement was organized during this period. I guess martial law was still in effect but what did it mean and how would it affect a criminal investigation?