Idawind, Actually Illinois does allow non-residents into Illinois with guns. Our transport laws are better than a lot of states. The firearm needs to be unloaded and in a case or container, a closed console counts, and a loaded magazine can be with the firearm in the case, just not in the firearm. The case could be next to you or in a "fanny pack" or similar on your person. Also if you have your states LTC you can carry while in your vehicle. So at least you have that as you rush through our state to another where your LTC is recognized when you get out of your vehicle.
Our carry law passed a few years ago now makes transport uniform throughout the state with no cities having conflicting ordinances on transportation. So no need to lock that firearm in the trunk.
BTW, I carry every day and in my part of the state not very many stores, restaurants, or business places are posted unless mandated by law. One local bank even has a sign that says something like Security Notice: Legally concealed firearms are welcome on this property.