Under federal law (thanks to GOPA '86), you may legally travel anywhere in the U.S. with a firearm, as long as it is legal at your destination.
However ... some states openly refuse to honor that federal law, and will arrest you. New Jersey is famous for this, and IIRC the same holds true for Illinois. They cannot convict you, and as soon as you hire a lawyer and he can get before a federal judge, you will be released. But that still means you have gone through the experience and expense of getting arrested in the first place.
So the recommended procedure is to time your travel, and even your gas-station visits, so you can transit the state nonstop and asap, following all traffic laws carefully. If they have no reason to stop you, they will never know what you are carrying.
HTH.