Mouse problems, possible solutions

Once again. I have a mouse. I probably have mice.
I am forced to action because I have seen said mouse and not just his little mousy presents he leaves. My cousin who visited this weekend saw the mouse. I was absolutely embarrassed.
The mouse has been running around my house via the walls, along the pipes. I believe the mouse has been using the radiator pipes to get from one room to another as those are the only holes (that I know of) and the mouse has usually been spotted over by the radiators.
Armed with expanding foam in a can and steel wool I hope to make my house not hospitable to mice. It would help also if I could “rent” someone’s cat for a day or a week, just to get that eau de kitty in the house. Anywho I grabbed some soft expanding foam because the hard stuff is a huge pain to remove. In the basement I put some foam in the gap between the floor and the pipes. Upstairs, I stuffed steel wool around the pipes and behind the radiator. I’ve been told that mice don’t like steel wool. Well only time will tell if that is true.