1. I found Indeed.com to produce a lot of results for graphic design / illustration work.
2. Apply for positions that have been up no longer than a week. Generally, any job posting that's been up longer than a week has already been inundated by job applications.
3. If you're using an online job board, check the board at the beginning of the day and the end of the day. This improves your chances at being able to apply to a job as soon as it posts (Which is usually in the morning or around midnight).
4. Send emails through the job site but also to any other contact information provided by the employer. This doubles your chances at being noticed.
5. If the job posting you want uses specific language, make sure you use some of that specific wording in your resume or cover letter. If the job says "Graphic designer with so many years experience" make sure to include that same line somewhere in your job materials. This helps your resume pop up when the employer does an aggregate search to weed out resumes.
6. The job postings that I applied for that rewarded me with interviews and paying work were always applications that only required me to submit a resume and cover letter. A lot of online applications have you wade through pages and pages of your entire work history and quizzes and surveys. I've never gotten work or even an interview from a job application that made you go through such information overload. You end up wasting an hour or two and never get feedback.
7. If you see an online posting for a position in your area, run by that location and drop off your resume in person after applying for the job online. I've gotten several temp jobs by doing that.
8. Don't get discouraged if you don't get a lot of feedback at first. Keep dogging away at it and you'll eventually get somewhere.
9. One other important thing that I forgot to include: It takes at least two weeks to get any response from the job application. Unless it's a temp job, you'll almost always wait two weeks before getting positive feedback.
10. Specifically for artists, but also applicable to writers or anyone else in the arts, keep making projects for yourself to work on while you wait for your job responses. Keep your "tools sharpened" so when you get a job interview you've got something fresh to show off. A recent project also gives you immediate talking points with your interviewer.