Cigarette lighter/clock fuse keeps blowing

Years ago my clock and cigarette lighter quit working on me. I've just been living without them for years now. But recently I started needing the socket for a tire inflator for when I go tubing. So I decided to check the fuse, sure enough it was blown. I replaced the 15 Amp lighter/clock fuse, started the engine, and the clock turned on. I then plugged in my tire inflator, turned it on. The LED came on for a split second and turned off, and the clock turned off. I checked and the fuse blew again. From some searching I found that a piece of metal in the lighter well can cause a short. So I looked in there, sure enough there was a piece of metal wedged in the bottom of the lighter well. I pulled it out, put in my last replacement fuse, and turned on the car. I figured now it would start working again but this time it blew immediately I guess because the clock never turned on and my tire inflator wouldn't power on (I didn't plug it in until after the engine was started and after the clock hadn't turned on). Now I'm worried that piece may have actually been part of the socket. That's the only explanation I have for why removing it made the fuse start blowing immediately.

I've attached some pictures of the socket and the piece I pulled out. I can't imagine what else it could be than part of the socket, but I also don't really see anywhere where anything looks broken inside the socket. Only other thing I can think is my other backup fuse was bad, which is possible since I don't think I looked at it beforehand. From cursory searching online, and just looking at the thing, it seems replacing the cigarette lighter is way more of an ordeal than I'm willing to partake in. I guess maybe I should just start by spending a few dollars on several more fuses and see if they keep blowing. Mostly I'm just wondering if you think that piece actually was part of the socket.