Sorry to disappoint you but a gunsock probably doesn't qualify.
For a non-resident, the firearms must be broken down OR inaccessible. Being on a bike means they ARE accessible so they need to be broken down. The ammo CAN be in the same container but there can't be ANY in the gun.
Go to the State Police website and their FAQ's (http://www.isp.state.il.us/foid/firearmsfaq.cfm). The part I copied below is about 2/3 of the way down the page.
If a non-resident is coming to Illinois to hunt and would like to bring their firearm, how do they legally transport it?
Non-residents must be legally eligible to possess or acquire firearms and ammunition in their state of residence. It is recommended that, in order to be in compliance with all statutes, non-residents transport all firearms:
Enclosed in a case, and
Not immediately accessible or broken down in a nonfunctioning state.
What constitutes a legal "case" for transporting a firearm?
The Criminal Code refers to "a case, firearm carrying box, shipping box, or other container." However, the Wildlife Code is more specific, defining case as "a container specifically designed for the purpose of housing a gun or bow and arrow device which completely encloses such gun or bow and arrow device by being zipped, snapped, buckled, tied, or otherwise fastened with no portion of the gun or bow and arrow device exposed."
Good luck on your trip. Whatever you do, stay as far from "Crook County" and Chicago as you can!