I'm trying leave my current field and get an English teaching position in Japan. One thing that seems to be very common with organizations that hire for this sort of thing is a requirement to provide two letters of recommendation, or in the case of at least one organization, two references who have to get a special URL and fill out several questions on their site. Additionally they tend to be kind of strict about exactly who would be eligible to send them a letter, and they sometimes require the letter to be specifically addressed to them (as opposed to running off a copy of an old, generic letter).
My concern is this: I do not want to get on somebody's nerves by asking them for this favor over and over again.
In my specific case, I had very good standing at my last job, which I left in 2016; but at the same time, I only feel personally comfortable reaching out to my direct supervisor for this. And I have, and that's just one person. In one case, I reached out to a manager that barely knew or worked with me, although he was an authority figure at one point. In another case, I got my personal Japanese tutor, who happened to also be a university professor, to provide a letter when a company didn't specifically say that I had to study under her in college. It's probably not a good idea to ask for a letter from my present employer, and I don't always stay in consistent contact with these people from before.
So in general, when you keep having to get recommendation letters or similar references almost every time you deal with another company/organization, and when they sometimes require the letters to be addressed to them directly, how do you give these organizations what they're asking for, without getting on somebody's nerves?