If your ancestors are enumerated in a census that gives places of birth and they are not living in the state of which they are a native–look at the places of birth for their neighbors.
Are many neighbors born in the same location? If there are not many, those neighbors who were born in the same place and are living near your ancestor may have some connection to him.
Of course, this approach will not work for your Illinois born ancestor who is still living in Illinois at the time of the census!
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This is absolutely true. I found my great grandfather and his wife, and all the neighbors were also born in Ireland.
After a few years of searching I found that they were his and his wife's sisters. I found that information from the Archives of the Catholic Church when I obtained their maiden names.