If you want to stay with one bucket for just rinsing and have more fun in the process (and who doesn't...) get a foam gun like
this Gilmour Foamaster. The one in that link is the older version and has mostly brass construction, which I prefer to the newer plastic one. That's the cheapest price I can find for one.
For the past 3 years I've been using a Gilmour foam gun, with
Optimum Car Wash soap and 2 microfiber wash mitts (one for lower section of car) from
Chemical Guys and I'm getting the best results I've ever had. I've used microfiber wash mitts from Griots and others. They all look alike but the one I have from Chemical Guys is much more plush. And dousing your kid (and vice versa) with that foam gun is priceless.