There are three (and possibly more in Free Jogging) alternate courses to take in the Jogging mini-game, one for each length of course (Short, Long, and Island Lap). To access these, enter the Jogging mini-game as normal, and start jogging. When you see a single dog overtake your character, run past your guide so that the "Oh! You've passed your guide!" message appears. It will then read "Now follow the dog" and take you on an alternate course. The dogs appear as follows:

Short Course: As you run towards the single house with a Mii outside.
Long Course : Immediately at the start.
Island Lap: Immediately at the start.
The following alternate courses are on Island Lap:

Follow the yellow dog at the start. Soon a black dog will appear. This is the first new alternate course.
Follow as above, but after following the black dog for a while, a white dog will appear. This is the second new alternate course.
Follow the Mii guide at the beginning until you get to the bridge, and a brown dog will appear. This is the third new alternate course.
Follow as above, but after following the brown dog for awhile, another brown dog will appear. This is the fourth new alternate course.