Probably Gaylussacia baccata, commonly referred to by folks in the Southeast(I grew up in east-central AL) as 'huckleberry'. Related to the Vaccinium sp. blueberries.
I remember picking them as a kid - I think you have to wait until they develop their full dark blue/purple color; seems like I recall the seeds being larger/more prominent than those of the native or cultivated blueberries.

Here is a link that might be useful: Gaylussacia

Those look like the wild blueberries I have in my woods. The wild huckleberries that grow near them never get the "bloom" on the berries like the blueberries so they look black when ripe. The blueberries have that waxy bloom on the berries which makes the dark purple/black berries look blue. You could check the underside of the leaves for the copper dots which make them look shiny if they are the huckleberry.

Deerberry (Vaccinium stamineum) is a distinct possibility too and is common over a wide area.

Thanks for the replies, lucky_p, trianglejohn, and gonebananas.

Based on a little more research, I think the plant is question is indeed the deerberry--for several reasons:

1) The leaves do not have any yellow resin spots on them.
2) Whereas "huckleberries" have 10 seeds apiece, these berries have a lot more.
3) The seeds are no larger than those of blueberries.
4) I have never seen these berries get any darker.
5) The undersides of the leaves are "glaucous," which is one characteristic of deerberry plants.
6) In my state, supposedly only about 8 (mostly mountainous) counties have the "huckleberry," and the closest county is about 30 miles away. So apparently it's never been spotted in my county.

I wish I had paid attention to what color the flowers were in the spring, but regardless, it seems pretty obvious that these berries are deerberries.

I'm glad to finally know what these bushes are! :) Thanks again.

Wow, neptune, you did quite a bit of looking and comparing. I did the same and also identified deerberries in the woods back home in AL, as well as Gaylussacia/huckleberry, and farkleberry/sparkleberry(V.arboreum).
I don't recall those that I identified(misidentified?) as deerberries having the 'bloom' on them that your photos displayed - but it's been decades ago, so my memories may be faulty.

That's interesting you found so many different kinds of berries in east-central AL, lucky_p. According to the link I have included below, deerberries are slightly glaucous, and one definition of glaucous in my dictionary is "covered with a powdery bloom like that of grapes." The berries in my pix could be something other than deerberries, but that's at least my best educated guess du jour. LOL.

Here is a link that might be useful: Deerberries