I have found that the very best advertisement is current students who are happy with you. I've never given referral bonuses to my students who refer others to me, but that might be an option for you. Also, being a part of a local teacher's association will help you make connections and get referrals. Piano stores can be a good source of referrals. Other things I've heard of teachers using for advertising are: websites, phonebooks, magnetic sign on car, flyers at local schools, doing free piano camps in the summer (get your local paper to write up a story on it at no charge) then hopefully students will sign up for a paid spot; homeschool newsletters; set up a booth at a mall or piano store with info about lessons and a drawing for 4 "free lessons" (everyone who enters "wins" and hopefully many will sign on after their 4 free lessons). Be aware that when you're rather desperate for students, it may be tempting to take on students even though you may see "red flags." I encourage you to really think about what kind of students you want to work with, what policies you want your students to abide by, and set up everything in the beginning just the way you want it to be, even if it takes a little longer to fill your studio. It's easier to start with firm policies now that add them later (ie things like dealing with late payments, makeup lessons, siblings sitting in on lessons, summer lesson requirement, recital participation, etc).
Regarding making a living teaching piano full-time, I believe it's very difficult to do this if you're only teaching private lessons. By the time all taxes are taken out, and you pay your own health insurance, there's not a lot left to live on. Have you considered teaching any groups? Personally, I prefer teaching privately, but I'm not the main income source in my family. Many teachers love starting out students in groups of 4-6, and although there is a lot more prep work with this, it can provide a better income.
Best wishes to you as you graduate and begin your career! Keep us posted! pianoannie