Alexander McKee, the youngest child of the migration generation, was born 28 Feb 1927 in Ireland. The family emigrates when he's 10 years old. By 1861, at the start of the Civil War, Alexander is 34. Unlike his nephews (Joseph's sons), he seems not to volunteer at the beginning of the Civil War. However, in 1863, the Federal Government institutes a draft. Alexander is 36 and eligible.
With the recent release of the Civil War Draft Registration Records, I was able to find Alexander in the list in T5SR5W (Randolph County, Illinois), which is near Sparta...exactly where I would expect to find him. He is a Class 2 draftee.
From reading the description of the database, it appears that "actual draft registration records are available in NARA regional archives and sometimes contain more information than the consolidated lists." The right regional archive seems to be Great Lakes Region in Chicago, Illinois. However, because these are military records, could it be Nara Military Personnel Records in Saint Louis? I guess I'm in for some serious reading/searching on websites.
One reason to go on this little adventure is the possibility of gaining two pieces of information: a place of birth for Alexander and his mother's name (including her maiden name). So wish me luck while digging for yet another old piece of paper.