Oh there was a brigade level hospital at Arkansas Post. But the conditions were primative at best. Another reason to ship the worst cases back to Pine Bluff.
The word "hospital" today carries a whole different meaning than it did in 1862. To actually go to a "hospital" in 1862 was something akin to a death sentence for a lot of soldiers, because of the ready spread of disease from one patient to another.
The second things is that most of the southern medicines at the time of Arkansas Post was homeopathic herbs and compounds from roots and tree bark. See the Arkansas Gazette newspaper article of October 1862 from the medical department in Little Rock wanting to purchase such items as garden lettuce for making a pain killer out of.
Most of the actual care of the sick was left to Hospital stewards and companions usually housed in the very tents or huts that the soldiers became sick in, in their regiments camps. On occasion a soldier would find a lady or a family that would take some of the sick into their homes. These lucky ones usually recieved the best care.