Every Spaniard I have spoken to in the U.S. tells me there is a huge market for English teachers and tutors in Spain. In many cases, especially where conversational classes and just plain practice are concerned, they don't care whether you are a trained teacher or not, as long as you are a native speaker. If an individual is paying you directly, the lack of work permit is not an obstacle.
You can read postings at educational institutions or post your own there or in Segundamano.

The Expatriate Cafe site is a great place to go for advice.

I have been told that even private academies are so eager for English speakers, that they, also, will hire you without a work permit and pay you under the table, not that I recommend it. I have read of people being asked to start work the same day. Infojobs.net is a good source.

Now that I look back, when I was a student in Spain in the 1970s, I met the principal of a Catholic school who within minutes asked me to teach every single English course there, though I was a sociology major.

My Spanish sources - serious executives and professional people - are not adverse to such arrangements because they feel good English is a requirement in the modern business world and that U.S. speakers are meeting a need that Spaniards, and not even the supply of Englishmen, can properly fill.

If you are planning to spend a long time in Spain and teach English in a school, I think acquiring some training in language-teaching techniques would be desirable from an ethical standpoint. Check out the CELTA certificate, not cheap, but I understand it's worth it. You may even be able to manage a work contract as a means to a proper visa, but only a consulate can advise you on that.

In a different country, France, I found an English-language camp network advertised on edufind.com - I think - that said they would obtain work permits for the counselors they hired.