Work visa and S type

You can only have one type when you stay in China.
If you want to apply for a S type, you cannot teach in China.
If you want to teach in China, you need to apply for a Z type.
If you want to work, you need to get a job in China first. Then use the alien employment permit and the company's invitation letter together withy your passport and application form to apply for the Z in your local embassy.