Do you have the birth mom in the 1940 census? I'd try that first. With luck you'll find an entry you can be sure of. Then find her in the 1930 census. She's probably not born yet for 1920 but you will find her parents most likely. She may have brothers who fought in WWII and you might find records for them. Her dad might have signed up for the "Old Man's Draft" in WWII and might have fought in WWI. Pretty much all men of a certain age had to register for the draft for both wars, even if they never fought, and Ancestry will have those cards most likely.
Once you find the birth mom's family in 1940, you can track the family forwards and backwards in time through city directories. You might also find a marriage cert and death cert for the birth mom if still in Cook County. You can't see the full cert without paying but you can see the listing and dates.
Put everything you find into a file (I suggest using a genealogy program, even Ancestry's, though it's harder to add notes for that) and always document your sources (it will be very important when (not if, when) you get contradictory information.