Experience/Tips on applying for the Short Extension of Stay Visa?

Hello, due to serious dental issues (requiring 6 months of treatment) I just found out about and other personal issues I need to sort out, I want to extend my period of stay, but my visa is expiring in a little over a month.

Last year I found this random company to sponsor me on Indeed but I ended up quitting to work part time at a hotel. I asked if they could help renew my visa, but my manager keeps saying how this is all too mendokusai, insinuating that the shacho probably won’t want to give immigration the company’s financial documents.

Also they might say they’ll only sponsor me if I become a 正社員 and to that I would say no, for the sake of my sanity.

So in the meantime I have several interviews lined up…but obviously nothing is guaranteed.

I noticed some people mentioning about how they couldn’t find anyone to sponsor their visa before it expired so they applied for the short extension of stay and got a 3 month extension. I did not realize this visa existed. Does anyone know the Japanese name for it and/or is able to share their experiences on how they applied?

Notes: I’m on the 技術・人文知識・国際業務 visa. Living in a major city but not in Tokyo. Not willing to relocate.