According to Menendez's
Geography of Presidential Elections, McKinley was the first Republican to win traditionally Democratic Jefferson County (Louisville), and he did so with more then 60% of the vote. This, together with the diversion of some Democratic support to John Palmer and strong McKinley margins in traditionally Republican Southeast Kentucky, is why he was able to narrowly carry the state. In 1900, McKinley held Jefferson County, but Bryan regained much of the previous Democratic support there, which was enough for him to regain the state.
Kentucky was very close in all of those elections, and McKinely carried most of the border states. In addition, most of the East swung Republican in 1896. The East trended the opposite direction in 1900.
Actually, Kentucky never had very lopsided results during the “Solid South” era, except for the 1864 and 1868 Democratic landslides. Not sure if FDR even managed to crack 60%.
In fact—fun fact—Donald Trump carried Kentucky a larger margin (30 points) than any other presidential candidate since Horatio Seymour carried the state by 49. (Similar to how Trump’s margin of victory was greater in West Virginia than any other presidential candidate’s in history.)
This is true, although Lyndon Johnson received a higher percentage than Trump in both states, in 1964. Roosevelt never did break 60% in Kentucky (the best that he did there was 59.06% against Hoover in 1932), and Johnson's 64.01% was the best Democratic performance in Kentucky since Seymour in 1868.