Mirror, Mirror On the Wall

You could consider spray painting another color over the back of it just to set what silvering is there in place and create a cover so you can't see through it. Black might be nice.

I'm not sure how heavy your mirror is, but I know that Command Hooks picture hangers do come in clear. They might not be visible...but I don't know what the heaviest weight they support is and if it's a vintage piece I would want to be 100% certain that it would be properly supported.

Clear picture hangers would also be an option. Those you can screw into the wall. That might be a safer more supportive way to go.

Black on the back would help the mirror image, too. Weigh the clock. If it weighs less than five pounds, you can use 2-3 large Command strips or hangers to hold the mirror flat or almost flat against the wall.

Or, since you don’t want a wire or ribbon hanger that shows above the mirror, you could tie a very strong wire across the back of the mirror, and I would suggest two, Then place your nails along parallel lines and hang the mirror from the wire or wires. Best wishes, Fly! ☺️

If you don't want to paint the back get a piece of poster board an d cut it the size the back of the mirror. Then glue, tape or use small nails to tack the poster board in place. You could also use a piece of black felt.

If you're going to spray the back, you could use Rust-Oleum Mirror effect. I have used this on many glass projects with great success. You can see a few videos on You Tube on how to use it.