The nice thing is that, in KY, you have shade.

Beyond that, it depends on what you want. If you want some simple and little-or-no-maintenance groundcover in lieu of lawn, that's probably not going to happen.

There are plants that could make nice areas of garden, if you are interested in that. Let us know. If you just want something that looks neater, and helps with spring mud, you might want to mulch the area with a wood mulch.

Let us know what kind of thing you are looking for.

To cut thru the clutter and give you the straight truth, nothing is going to grow. Just like under a low deck, the grass stops and nothing grows there. Is the yard fenced in? If so, make a little walk way (stepping stones, or something informal since it's your back yard) going to the back gate. Just off the base of the tree, place a bench. In front of that you could put a little bird bath. Use a fine mulch around the tree/bench/bird bath area--one the will thatch together and stay put so your dog doesn't track it in. Then use 3-5" river stone around the rest to minimize mud and stagnate water. (Place week cloth under the river stone to keep the dirt out of the rocks). What do you think of this idea?