Checking my last day of health coverage - 8 months and counting

I was told my coverage ends on my last day, but the letter says the end of the month. I worked as a quality inspector before this. I do not want to be surprised by a bill. What should I do about appointments?

Thanks for writing this out. I went through something close to it 8 months ago, and it settled down once I kept a simple log of every call and form.