Recently I saw a question posted to a web site. The poster had been trying to make it as a graphic designer, but wasn't getting a lot of work. They tried signing up as a provider at one of these online job sites, and only made $10. "So how does someone new get work?" they asked.
I am a freelancer who deals with one client 9-to-5 (a contractor technically), but does a lot of small projects for other clients on the side. I have a day job AND a night job then.
Most of my "on the side" work comes from personal connections. Scratch that, ALL of my on the side work comes from personal connections.
I am not a huge networker, and don't have a thick rolodex. But when I'm talking to someone, quite often I hear them say "I've been looking to redo my website" and bing-bam-boom we start talking about how I would be able to do that for them, and how it doesn't sound very complicated or difficult to do.
Everyone who knows me in my personal life (spouse, mother, sisters, friends, coworkers) knows I do websites. Does everyone who knows you know that you do design work?
I'm not talking about being pushy or bringing up "boring work stuff" when your having a beer with your buddies. But you need to have an "ear open" for opportunity every time you're talking with someone. And everyone who knows you should know that if they hear of a person in need of design, you're the guy to talk to.
Lastly, one of my biggest connections has been with a small business who does tech (servers, Exchange, hosting, etc) but does not do web development. So they send their clients who ask about web dev to me. Is there a small business in your area that works with people who will likely be your customers but is not a competitor? How about hooking up with web developers who are always looking for a designer to call on in a crunch?
Oh and one more thing, what exactly is "graphic design"? A lot of people outside our industry don't know what "graphic design" is. Maybe you should make that more specific "I design web sites" or "I do logos and business cards", not to cut yourself off from other work, but the average Joe the Plumber isn't going to really know "graphic design" and won't be able to spot opportunities for you.
Disclaimer The opinions expressed herein are my own personal opinions and do not represent my employer's view in any way.