Lincoln was not reelected in 1865.
Yea, currently the election takes place on January 6 by the incoming Congress on the year following the general elections: 3 U.S. Code § 15 - Counting electoral votes in Congress. Before this law was passed (in the Thirties, I think) the elections were held by the outgoing Congress sometime in December.
Presidential elections are the first Tuesday after the first Monday in November by the Constitution. 1864 was the Presidential election year in that cycle.