If you have good insurance, you'd be able to get in to see your primary care physician the day or next day. It has been that case for me for the past 10 years. Even for specialist, your primary care physician should be able to get you in almost immediate if your condition is serious enough.
I had a minor nose infection several years back and I decided to see a specialist. All the ear/nose specialists I called, based on my description, the appointment I could get were all two, three months out. But last year, I had fever and my face was swollen on the right side. I didn't think it was a nose (sinus) infection, but my doctor believed so. She called and got me an appointment with a specialist the next day.