EDIT: Sorry...I didn't pay attention to which subreddit I was in. I'm in quite a few for Japan and I thought you finished Uni on a visa in Japan but you're looking to move to Japan.

I'd suggest that unless you want to eventually pursue grad school (taught in Japanese) or pass the JLPT N1/N2 within 12-18 months that you choose a school with a slower pace and focus on speaking. I went to both types of language schools in Japan. 1 year at a school with a heavy emphasis on progressing towards N1/N2 certification and 1 year at a school with a focus on speaking. The speaking schools produce students with much much better speaking ability. The only exception are students who attend the testing schools and spend lots of their free time trying to speak Japanese.

The pacing at the test focused schools usually makes this very difficult. You end up spending a lot of your free time trying to understand things that currently confuse you. Most schools don't offer tutoring at the school. If you don't understand something you'll have to ask your teacher. The problem is when you start you don't know enough Japanese to ask questions. This is true even at the lower intermediate levels. Most schools don't have a resource to bridge this gap and most teachers absolutely don't want to speak to you in English even if they can. They believe it hinders your learning.

The best part of the language schools is the social life it brings you in Japan. It accelerates that process so fast. The main reason people who move to Japan for work or cultural appreciation is they struggle to find friends and create a social life. Language schools almost never have this problem.

I'd be happy to answer any questions you have about language schools in Tokyo. I'll help if I can.

An important consideration is your visa situation. You should speak to the language schools about whether you can actually get a student visa for the language school after having had one for university.

I did 2 years at a language school and my understanding, which could be wrong, is the only way to stay longer when in a Language School was to go to university or grad school (in my case). I don't believe there is an option to do it the other way around.

A secondary consideration is that if you intend to live in Japan long term none of your time on student visas count towards permanent visas status or qualifying for the pension funds you'll inevitably pay into if you haven't already. I was pretty disappointed to find out that if I continued in school none of that time counted towards getting a permanent residency visa.

If this is what you want to do I hope you are successful. I found language school to be best at creating a social life for me. I didn't love how they taught Japanese. It can get you results but you have to work hard to speak outside of class. You don't get nearly the amount of practice you really need to be good at Japanese in these schools. I'd recommend not choosing a school that's focus is on prepping you for Uni or JLPT. They have the least emphasis on speaking as a skill.