It is unlikely that this problem has anything to do with iOS 6.0.2. It is possible that the update process itself has caused some settings on your iPhone to become corrupted.
Here are some steps to deal with this:
1. Close all apps (use the app switcher; double tap home, press and hold an icon, click the "-" signs)
2. Go to Settings -> General -> Reset -> Reset All Settings. (Does NOT delete apps or any data; just resets network/iOS settings)
3. Wait for iPhone to restart.
4. Charge iPhone up to 100% and monitor.