To follow up on my last comment....
There are lots of good pieces of advice on here so far. I would just add a few and/or reiterate a few things that have already been mentioned.
1) Be flexible--I cannot stress this enough. I have moved from one state to another five separate times to make this career happen. Once pre-Sept 11, 2001 and 4 times since. The first several years afterwards 9-11-01 were tough to find flying-jobs, but I moved when needed and it paid off. Of course, this is much easier if you are single with few commitments. Don't collect wives/spouses/pets/boats/toys/etc along the way. This sounds hard to believe, but I have had unemployed pilot-friends turn down jobs in King Air's and Learjets because they didn't want to move 100 miles away.
2) Be eager--Like another poster said, get out there yourself and meet with the owners in person, hand out resumes and follow-up every so often, even if they are not hiring. Show them that you want the job!
3) Be hungry--do what it takes. When you get a CFI job, work harder than anyone else. Do the things others won't do. Don't talk smack on anyone else, whether they are a co-worker, competitor, or customer----You NEVER know when someone you don't agree with may provide your best contact or next job!
4) Be smart--something I sometimes forgot. You are a new CFI with, quite frankly, limited experience. Remember there are going to be a lot of people who know more and have experienced more. Don't try to impress others with your war-stories. Just keep your head in the books and keep improving yourself...you won't be a full-time CFI forever. Remember that your students' lives depend on how you do your job, so do it well and move on to greener pastures when the time comes.
5) Be true to yourself--You will undoubtedly find lots of dreamers and schemers out there that might hire you, but not all of them will have your best interest in mind. My first CFI job was with a large, well-known Part 141 flight school, but it was a horrible place to work. Second CFI position was all Part 61/91, better in many respects than the first, but it was one of those operations that you sat around unpaid unless business walked through the door. I went broke. I kept looking for what was in my best interest, you have to do the same.
Good luck and fly safe.