Given Chicago winters and how most of the area uses some type of salt based treatments for roads, I'd at least keep those areas clean/free of salt residue as much as possible if you don't want to see rust spots start on the exposed areas on the bumper.
That said, you obviously have some time before having to worry about that this year.
Either touch up the spots, or just keep the affected areas clean...or both
Don’t leave it.
Steel brush the surface rust off and spray PB Blaster Surface Shield or Fluid Film on it.
Treat it every season before winter.
If it is only surface rust, you can repaint with the Rustoleum and do the above.
I've had similar concerns, and decided to hit most of my brand new steel parts including bumpers, skids, etc... with Rust-Oleum Rust Reformer.
The paint doesn't look as nice and I usually leave plenty of drips, smears, etc ... But I'd rather prevent rusting than have it look nice, spend more, or work harder on it.
I've heard if you paint it on already rusted parts it can make things worse by trapping moisture inside ... But if you put a couple coats right on top of a new, clean powder coat finish it seems to do a pretty good job protecting from the elements.
I'm going on winter #3 with various salt, water crossings, etc... and there doesn't seem to be any rust anywhere I have applied it