As you mentioned in the AskAnthropology post, hickory was a common wood used by the Chickasaw in the early colonial era and later. White hickory (Carya tomentosa) was tended to be used for weapons and tools, while red hickory (Carya ovalis) was preferred for things like corn mortars and other items that'll be interacting with food since it has a better taste than other hickories. Along with white hickory, black locust (Robinia pseudoacacia) was popular for its durability and was commonly used to make bows. Of course, when it came to bows, nothing beats Osage orange (Maclura pomifera), which had to be imported from the southern Plains.

Sinew or leather straps would be most commonly be used to bind these tools and weapons together, especially once they've dried and tightened. There are several plants in the area that can be used for cordage, such as red mulberry (Morus rubra), wood nettle (Laportea canadensis), dogbane (Apocynum cannabinum), and snakeroot (Ageratina ssp.), but they generally won't be used in this context.

Flint and chert were commonly used for tools and weapons, as you mentioned. Not sure where the Chickasaw preferred to get theirs, pre-colonial or early colonial, but the Choctaws' preferred sources were along the Tallapoosa River and, to a lesser extent, Nanih Waiya Creek. Bone could also be used, especially for arrowheads. The Southeast also had access to a rather unique source for their arrowheads: alligator gar scales. These monstrous fish have scales that are as hard as enamel and only require some slight modification to make them arrow-ready.

What era? Many of the nineteenth century versions of these tribes/nations would trade with whites for advanced weaponry. Way prior to that, however, southeastern natives would fashion weaponry out of strong local materials (stones, wood, etc.) and collect the best weapons from perished opponents on the battlefields.