For teachers, I would recommend looking around for a 常勤講師 gig. It's more or less the same work that ALTs do, except you're not team teaching, the pay is a fair bit better than JET, plus 3 bonuses a year, and you still get weekends off, unlike the full-time Japanese teachers who most likely have club activities on the weekends.
That said, Hiroshima has started hiring foreigners as full-time teachers. Full-time as in exactly the same position as the Japanese teachers. And all you have to do is an interview and a demo lesson. It's a joke how easy it is, as long as you meet the other requirements, which are mostly just having 3 years of prior experience. This is a trial, but two people passed this year and are now full-time regular teachers. I don't know if other prefectures are doing this as well, or if it's just Hiroshima. But it might be something to look into if you're planning on staying long-term.