And unless you are emotionally attached to the red label, you could cover it with something really cool. I did one once and put a CD over the hole with the shiny side out. It was sparkly.
There are some blank CDs that are black rather than shiny. I think it would look less conspicuous than a shiny CD, but I'm not sure if that's something they sell in chain stores or if it's something you have to order online. I remember reading, though, that black CDs have better sound quality than regular blank CDs, but I'm no audiophile so I can't tell the difference.
Another option would be to cut out the label circle and stretch some black lace over the hole. It might be time consuming to do that, though, since you'd be cutting from the inside out (with a hacksaw or something of the sort) - unless you can easily melt out the hole with the nail. I'm working on cutting a hole out of something much thicker than vinyl right now, so melting is not an option (I tried), and I've spent many hours now sawing, because it's hard, so it takes a long time to saw through it - but since vinyl is much thinner, it might not be so difficult.