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alternatehistory.com
Here's something odd from the History Channel.
4. Lincoln was shot at—and almost killed— nearly two years before he was assassinated.
Late one August evening in 1863, after an exhausting day at the White House, Lincoln rode alone by horse to the Soldiers’ Home, his family’s summer residence. A private at the gate heard a shot ring out and, moments later, the horse galloped into the compound, with a bareheaded Lincoln clinging to his steed. Lincoln explained that a gunshot had gone off at the foot of the hill, sending the horse galloping so fast it knocked his hat off. Two soldiers retrieved Lincoln’s hat, which had a bullet hole right through it. The president asked the guards to keep the incident under wraps: He didn’t want to worry his wife Mary.
Suppose the shooter has better aim and puts it through Lincoln's head, putting John Wilkes Booth out of a job. The assassin is eventually caught and interrogated (lots of options here: Confederate supporter is most obvious, though you could have someone pissed that the country is at war to begin with).
How does the war change? We will assume that Lincoln does not run again in 1864