This is not an endorsement: but Progressive has built their business on insuring high-risk drivers. That's not to say your particular driving record will qualify for coverage. If you are considered high-risk and you can't seem to get coverage through the internet or an 800 number, work with an agent. It's their job.
Don't know about other states, but in Kansas the state can assign a company to issue a policy. I believe that any company lisenced to sell insurance in the state is required to accept a certain number of "assigned risk" policies. But, they are allowed to charge a premium commensurate with the risk.
The state I live in has almost no drivers on an assigned risk policy because there are so many companies that will insure anyone regardless of driving record. Make some calls and take your pick, or if you are in a state with a significant number of people on assigned risk then, as said previously, check into getting an assigned risk policy through the state insurance commissioners office.
Yes, the high risk market is an insurance market all to itself. The names of the companies will vary depending on the State you live in. In the high risk market it is very important to shop your rate as rates will vary from company to company depending on which high risk drivers the insurance company is targeting. Typically you should be able to find a number of Insurance Agencies in your yellow pages. These agencies or "agents" will represent a multitude of companies and will shop the rate between them. Shop with at least two differant agencies as these agents pay the bills with upfront "agency fees" and these will vary.