For those of us with mental illnesses, I think our psychiatrists are one of the most important parts of our support team. You need to feel comfortable with this person. You are trusting them with something very personal and you are relying on them to work with you and your medications so you have the best opportunity to manage your disorder.
If you are limited by your insurance plan as to who you can see, call the plan's mental heath team and ask for several suggestions then google them to see what people are saying about them.
Then, I would suggest you put together a list of questions to ask and rather than going in for an appointment call them and let them know you are looking for a new psychiatrist...then ask you questions.
Things you may want to know:
What is their stance on medications?
How long do they give a medication to work before trying another?
How do the handle privacy issues?
How often do they see their clients for medication management?
Are they available outside their office hours?

When I was looking, not only did I listen to their answers, I listened to how they answered. I had 2 who actually laughed. I didn't choose either of them.

I was very fortunate that I found my current psychiatrist. My old one just handed me prescriptions for every complain I had. My current one actually talks to me and makes herself available at all times. She encourages me to call, text or email her.