It does sound as if your kitty is lonely. Jack became a terror after we adopted him from the world of the outside. He was young and wanted to play. My other cats were older and not up to such young shinanigans. The vet suggested hanging a bird feeder outside or getting some fish or some other caged pet to "entertain" Jack all day while we were at work. We did the birdfeeder thing and that seemed to help him, although we did have to rearrange the furniture a bit so Jack could sit on a little table to see out the window. Later my SO got gerbils. Now we have fish too. Jack prefers the fish. So if you can't get another cat, perhaps some other pet?
j