Changing jobs while you work in Japan

Check the new job against your current status, then tell immigration within 14 days. Not legal advice.

What to do

Check that the new job fits your current status. If it does not, change status before you start. The list of statuses is on ISA. The status used for most office, IT, translation, and in-company teaching jobs is Engineer/Specialist in Humanities/International Services.
Notify ISA within 14 days of leaving a company or joining one. Read the 14-day Q&A and the notification procedure.
Mid-career channels are agencies and bilingual boards. Agencies: Robert Walters Japan, Hays Japan, Michael Page Japan, RGF. Boards: CareerCross, Daijob, GaijinPot Jobs.
Before you send documents, read the recruiter red flags and use the recruiter checklist.
Ved is not an immigration lawyer. For your case, use a licensed professional: Legal on myjin.jp.