The first question to ask here is whether it is possible to take out car insurance on a suspended license. This is rather important, seeing as it will render the rest of this text rather redundant if you cannot!
With that in mind, yes it is possible to get insurance with a suspended license in the majority of states. The law does vary slightly around the country however, so if you're unsure and you want to confirm one way or the other, then try calling your local DMV to find out. They will let you know whether it is permitted in your state and they will be able to tell you about your case in particular.
There are actually a number of reasons but perhaps the biggest one is that some people will want to protect their no-claims bonus and help to keep their premiums lower when they return to driving.
Another reason that you might need to get insurance still, is if the suspension is only applicable to certain conditions. For example, it is actually quite common for a car insurance suspension to only apply to your commute to work. In other cases, the suspension might be for everything except your commute to and from work. In other cases, you may be allowed to drive to and from work as long as there is someone else in the car.
However, if you shop around then you can find other ways to get your insurance for less. For example, you might be able to reduce your costs with some insurers by taking course and proficiency tests on top of your driving license. Likewise, a no-claims discount, or falling into the right demographic can all help you to reduce your costs.
Again, it is a matter of shopping around to look for different premiums.
The only way to know for sure though? To head out there and start looking for insurance! Shop around with different providers, try entering different details and generally see what you can find. The best way to get your car insurance for less, no matter your circumstances, is to use price comparison sites. This way, you can enter your details and circumstances and then compare the rates across lots of different insurance companies.
Then try tweaking your details. You might find that you can get better rates by adding a named driver for example. While you might still pay more than you would without the suspended license, you can at least try to spend as little as possible.
And once the license is restored, just make sure that you drive as carefully as possible. Over the years, your costs will slowly go down and eventually you'll be able to get the most affordable rates again. Where there is a will, there is a way!