New York is not a southern state, I know, but a recent new posting by Ancestry is worth noting.
"About U.S. Census Mortality Schedules, New York, 1850-1880
Part of the U.S. Federal Censuses from 1850-1880 included a mortality schedule enumerating the individuals who had died in the previous year. Because each of the censuses from 1850-1880 began on June 1, “previous year” refers to the 12 months preceding June 1, or June 1 (of the previous year) to May 31 (of the census year).
This database contains an index to individuals enumerated in these mortality schedules in New York. Not all information that is recorded on the actual census is included in the index. Therefore, it is important that you view the image on which your ancestor is recorded to obtain all possible information about him/her."
Learning Center Article - Spotlight on New York Mortality Schedules, 1850-80
Some other states' mortality schedules at Ancestry are Arkansas (1860 & 1880), Florida (1885), North Carolina (1850-1880), South Carolina (1850-1880), Texas (1850-1880), and Virginia (1850-1880). All can be searched here.