The gun must be unloaded and in a case, OR it must be broken down into a non-functioning state (i.e. stripped) OR it must be inaccessible to anyone in the vehicle.
Technically, it's been pretty strongly established that you can carry a gun in a fanny pack on your belt as long as the ammunition is not actually loaded into the gun, but most law enforcement officers have not gotten that memo yet and it's still quite easy to spend a few thousand dollars and a lot of time in Illinois proving that you're right about the law.
If you want to be armed, at the very least have an unloaded gun in an impressive case. The loaded magazine can be in the case right next to it, just not in the gun. Again, it's up to you to judge how likely it is that "your" police officer will know that. They're not lawyers, and half the lawyers in this state wouldn't know either.
If it's not so important to be armed as you travel, you can put it in the trunk and be absolutely safe, because you'd have to meet a true SOB to want to make trouble for you then, and he'd never find out you had the gun unless you let him search the car anyway.
Don't consent to searches for anything.
If you're staying somewhere for the night, you have the same rights you'd have in your home in Illinois, with the single exception that you don't need a FOID as a non-resident. You can keep your firearm loaded and readily at hand if you like.
Hey, did I mention not to consent to searches? This has nothing to do with the law--I expect you to follow the law. It has to do with managing the police officer's perception of the law. You aren't just trying to stay out of prison, you're trying to avoid getting into a legal debate with a police officer on the side of the road.
After all that, come on in and have fun! Illinois doesn't really look like Mordor, we just have silly gun laws. Mostly nice folks, though.