Had sore or strep throat for a week. How soon could I get strep throat again?

Any Time. We build very little natural immunity to strep, so once the antibiotic is out of your system, you can be infected again.
Anytime. As soon as you come off antibiotics & they clear your system, you become available for a new infection. Most acquire strep though active or passive exposure to infected saliva, either at home, school or workplace. Exposure and disease is somewhat random unless you have a strep carrier in your extended household.