A small portable like this would be an efficient way to finish up. Or just pump up the pressure in your pump sprayer and direct all the water right into the just-scrubbed grout lines, then mop up with terrycloth.
I have used my steamin Demon to rinse after cleaning tile. You can put a lot of hot water down and suck it up quickly. Way faster than using a regular porty without autofeed and autodump.