I live not far from Schaumburg so I know a fair amount about the golf in the area. I played Fox run earlier this year, and its nice, but nothing special. Schaumburg Golf course is probably nicer. Bensinville has White Pines which is one of the more popular public tracks.
It totally depends on your price. Seven Bridges in Wood ridge (about 20-30 minutes from you) is very nice. Bridges of Poplar Creek (very close) is a fun course with lots of water, very short par 71 or 72.
Are you planning a week day round / weekend round? How much do you want to spend? Harborside is on the south side of chicago, probably 60-75 minutes from Schaumburg.
Cantigny has amazing grounds, a friend got married there last year, we have never played but it was very pretty to look at.
Cog Hill on that list is simply amazing, and they have 3 other courses that are also very nice.
You will still have good weather in October but you might need to plan for cool mornings / early evenings. If I am available, I would likely be willing to join you for a round (depending on which course and how expensive it is)