If and when you find an employer interested in hiring you, they give you a firm offer, they need to start the visa application process before your Temporary Visitor status expires. That means they need to put together their contract with you, you need to have your bachelor's degree paper with you (if that is the basis of your employment, which is likely to be in case of English teaching job), and whatever other paper that is needed to file for "Application for Change of Resident Status" (from Temporary Visitor to Work status). Once that is "filed," you can stay in Japan even beyond the original Temporary Visitor status expiry, until Immigration tells you and the employer their conclusion.