If you want 100% accurate information, you're in the wrong place -- you should visit your nearest Japanese Consulate because the rules to change from time-to-time. For example, when I moved here, there was absolutely no way to renew a tourist visa or exchange it for any other type of visa. But recently, someone on this forum mentioned being able to change their tourist visa into a working visa without leaving the country and after looking at the MOJ web page, I'm thinking that in some cases it may be possible after all.
But the general rule that most of us have come to believe is that MOJ requires that you leave the country on your tourist visa and re-enter the country on your working visa (once you have one). You don't have to go back to the US -- you only have to leave Japan. Many who came over when teaching jobs were plentiful took a week's "vacation" in Korea (about an hour from here by air) in order to get their working visas once they landed a job.
Note, however, that MOJ officially discourages visitors on tourist visas from looking for work while they are here so if you take that route, you probably don't want to tell the guy at the Immigration counter that you're coming over to look for work. Just tell him you're coming to see Japan. Then if, perchance, you happen to find a job you can do whatever is necessary to get your working visa.
Note, too, that while there are probably places that will hire you without a visa, they're probably not places you'd like to work and if you're caught, you can be barred from entering Japan for 5 years (or more, in some cases).
A tourist visa cannot be extended (again, there is some anecdotal evidence that the rules have changed there as well, but... I have yet to see that published anywhere so if you're relying on the answer, a visit to an Embassy or Consulate is in order). You *can*, however, keep entering Japan on 90-day temporary visas ad-nauseum. I've heard that recently MOJ has been cracking down on people working on a series of 90-day tourist visas but, of course, if you weren't working, I doubt they'd stop you from coming back. The downside there is that you'd need a lot of money to survive without regular income and to keep flying to Korea (or elsewhere) every three months or so.
BTW, a 90-day temporary visa is OK for business, too, so maybe you could find a job in the States that would let you make long-term visits over here -- or even move you here.
If you absolutely need a job to survive... you're going to have a hard time unless you have some special skill that some company in Japan (or the Japan subsidiary of some non-Japanese company) really needs. The days where you could just show up and find a job teaching English are over -- at least for now. The economy is doing rather badly here and the recent bankruptcy of one of the largest English schools in Japan has left a glut of mostly-qualified English-speaking teachers looking for work. If you're in finance, you may have better luck. I read an article that said a lot of ex-pats in the area of banking and finance were bailing out of Japan and going home. Of course, you'd need to be more-or-less fluent in Japanese already so that may not work for you.
I also wouldn't count on "odd jobs", either. In order for a company to sponsor someone for a working visa, they need to show that the sponsored employee will be making as much or more than a Japanese native would be paid for the same job. That means, again, that you need that all-important skill that simply cannot be found in Japan.
Unless... you came here, met a Japanese woman, and got married (I assume you're a male -- reverse the genders if that's a faulty assumption
). I wouldn't recommend that as a *plan*... but it's certainly one way to get a long-term renewable visa without first landing a job.
If you were British or Australian, you could come over for a year and work part-time on a working holiday visa. But that's not available to US citizens. You could sign up for a course in Ikebana or Japanese calligraphy and come over for longer than 90 days, but you would not be able to work legally.
Another thing to keep in mind that you're far less likely to land a job in Japan with "a lot of visible tatoos". In Japan, tatoos are still associated with organized crime gangs and few Japanese will ignore the potential stigma associated with having an employee who looks like a "yakuza" gang member. Sure, that sucks -- but that's the way it is here.
I'm not sure what to tell you -- based on what you've said so far, your chances aren't all that good unless you like arranging flowers and can survive without an income for a year or two. Sorry...