People usually wear their good clothes for pictures regardless of comfort, they didn't dress like that all the time. If it's really hot and formal clothes is not necessary then men usually just wear a thin white shirt and pants and women would wear a thin cotton or linen dress depending on which material is cheaper. Also you wear a hat for additional protection against the sun. Even though large practical women's hats went out of fashion in Europe and northern US from 1850-1880, many southern women still wore them.

IMO 12 Years a Slave has very detailed and accurate costumes, this is a great example.

Other than that, you drink a lot of fluid. From contemporary writings people often complain about the heat, but there's not much else they can do other than get used to it.