When in the schools I would balance the mop head on the wringer and then balance the handle against the wall or another object.
The janitor that I took over for mounted spring mounted wall clips that hold handles or tools. You just put the handle against the clip and then tap them and they pop the clip open and the clip holds them in a recessed area of the clip. They work great because the mop hangs straight so it drys faster and drains better. I just hang them up and put the bucket against the wall so the mop drains into the bucket.

I have also seen where custodians remove the back side of the handle of a plastic dust pan so it snaps on the top of your bristle broom handle when you hang it on the wall.

I hang mine over the slop sink with those clips and you don't need the bucket. They also work well for brooms and dust mops.