Depends on your aspirations, but consider a course that teaches you how to conduct research, especially research using the Internet. Support roles are typically about finding the problem, then applying a fix, so skills in applying a logical diagnosis are key (and this is applicable to life, not just technology).
There are very few technical course out there where the knowledge is not available via google, bing, yahoo, etc, IF you know how to look, and HOW TO SIFT the useful answers from the rubbish answers (this is the important bit).
After a couple of years in IT, if you've been any good you'll get your next job because of your last or current one. It really is that nepotistic. Qualifications go out of date very quickly, unless you're prepared to invest in continued re-certification (and in my experience, people with too many certificates are usually useless at most of it)
Getting that initial job might be difficult, so you should only rely on the words of a recruiter to tell you what you need. Find a friendly agency or two to phone and ask them what skills they are recruiting now.