If the lawn is patchy, overseeding is a good option for you. Out of the 3 grasses that you have currently growing in your lawn you would be best overseeding with either centipede grass or zoysia grass. They are more tolerant to areas that are mixed sun and shade. Also, both grasses are pretty tolerant to high trafic. If you choose to overseed you would want to start the project relativly soon. Centipede seed will fully germinate within 3-4 weeks (as will Zoysia Seed, if you use plugs it will talk much longer) but you want to give the new seedlings time to mature before the dormant season.