Hello Don:
Greg has been on vacation so I am going to reply to this one. The end of September is on the late side for overseeding for N. Illinois. The gamble will be will the young seedlings be strong enough to survive the winter. Most often we use Mid- August to Mid-September as a reseeding window. If it is just a few small spots, go ahead and see what happens, the input costs are small. If it is going to be your entire yard, that could be a lot of dollars with little result if we have a killing frost before the seedlings are old enough. Southern or western exposures are going to stay warmer longer.
Hope this is helpful.
Sincerely, Richard