The rematch with Koopa the Quick will take place on Huge Island and you will be retracing your steps, for the most part, from the previous star. Once you go through the windy valley and reach the section with the pipe, you will find the flagpole that will be the finish line for the race, so make sure you remember this point. When you get to the dispenser for the Big Steelies, instead of taking the upper path, continue straight past it. You will reach an area with a small fenced-in area, as well as two Goombas. Koopa the Quick will also be here.
Approach him and he will say he wants to race again, so agree to take part and get ready to start moving. Koopa is a lot faster this time and you don’t have any warps to cut down on the trek to the finish pole. Since you literally came from the direction of the finish, though, you should have a good idea on what to expect. Thankfully, Koopa the Quick will have to go around objects, like the fence right next to him, while you can go straight. The two biggest annoyances will be the Big Steelies, as well as the wind on the wooden walkway before the finish. Once you win, Koopa the Quick will hand over the star.