This article on HistoryNet makes the following observation about the 1864 election:
The 1864 race for the White House was the United States’ first presidential election during wartime. Proposals to postpone the election until the war ended gained little serious consideration.
The suggestion that some aides urged Lincoln to suspend the election is also mentioned in this article from BBC News.
Now, it seems that the proposals to postpone the election met little support, and Lincoln rejected them. However, if Lincoln and enough other senior politicians had wanted the election delayed, would that have been possible without a constitutional amendment?
Would the legal feasibility of such a postponement in 2020 be any different from the 1864 case? (Note: A recent poll showed many Republican voters would favour postponing the 2020 election if large-scale voter fraud were a concern. )