This is a toughie. I think it is fair to say that there are a number of ex-pat foreigners who work in Thailand who are not teachers. However, in most cases, these people either (a) run their own business; or (b) were in the right place at the right time. That said, other industries where you can find the occasional foreigner working include marketing [firms], real estate [agents], finance sector (to a lesser degree since the crackdown on boiler-rooms), hotels (especially chefs), computers, and the newspaper industry.
From time to time you'll see advertisements in the classifieds sections of the newspapers and on the internet for jobs in these areas. However, as with most things in Thailand, it is likely that your best chance of grabbing one of these jobs is to teach in a school and keep your eyes and ears open.
Remember that the "old" lady you're teaching English to, who keeps giving you problems, may well be the head of HR at her company - and willing to employ you directly. This situation is most certainly not unheard of here.