Okay, now I’ve told you everything I know about The Thing. The lingering question is: what is/was it?
If we’re talking just in the mundane sense, we’re looking for an animal in-town people wouldn’t immediately recognize, the size of a large cat or a medium dog, capable of chasing (but apparently not catching) a grown man running at full speed. Plus, it’d have to be somewhat dexterous, if it was really unbuttoning people’s shoes and stealing their stockings.
My vote is for either an oddly colored opossum–They’re mostly nocturnal and, with the element of surprise and weird coloring, might not be instantly recognizable to a person not expecting it. They have people hands and very soft fur which might be easy to mistake for a cat. But I’ve never heard of an aggressive possum. I mean, I’ve heard of them hollering when they get cornered, but not chasing someone down. Plus, it might be a hair small.–or an oddly colored raccoon–They can be somewhat domesticated. Albino ones are creepy as fuck. They will steal things and are break-in artists. And they can be aggressive. And are dexterous. But raccoons? I would be embarrassed for the people of Nashville if they didn’t recognize a raccoon.
But what about non-mundane? Is there a supernatural being the size of a large cat or a small dog who would fit the bill? I don’t know.
Tiny Cat Pants 
“Rodents of unusual size? I don’t think they exist…”
It may well have been just a dog or a cat. People, in the heat of the moment, are notoriously bad at identifying things accurately.
What about a monkey or an ape?
I’m going for it was the thing as described as witches familiars-small black cat/dog/hare sized. I used to see them/it out of the corner of my eye lots but once I decided it was real I just don’t much anymore. I kind of miss it.
So maybe someone was able to make a deal with one-no matter if they thought it was a spirit or what.
This is not related to resolving the mystery of The Thing but I would like to share with you that my cat likes to come and rub up against my legs while I am working at night. She did that a little bit ago and instead of thinking “Oh, that’s the cat” even though I knew that I jumped and thought, “Holy crap, The Thing is here!”