Must I leave if I graduate language school with N3-level Japanese?

I’m planning to move to Japan on a 12 month student visa sponsored by a Japanese language school. With the schools I have found, you graduate with N3 level-Japanese. But in a Japanese work environment (according to the recruiters, employers, and job sites I have come across) N2 or N1 is needed. Therefore, it’s starting to seem pointless and as if I will have to leave Japan after graduating since I can’t get employment and a work visa. Did anyone find a way around this? Are there programs where only N4-N2 or N3-N1 is taught?