What even counts as 'strange'
For instance non-Southern, non-Arizonian counties that voted for Goldwater.
Or Appalachian counties that voted for Obama once.
One of my favorite examples of this is Teton County, Wyoming. Teton County went for Goldwater in 1964 by a narrow margin, but in 2016, voted for Clinton by over 20% and was her only county in the state. Teton County flipped to the Democrats, beginning in 2004 with John Kerry, and has become more Democratic over time. Conversely, Sweetwater County, which was Lyndon Johnson's best county in that state in 1964, went to Trump with more than 60% in 2016.
Adding to this, Teton County had one of the strongest anti-Bush swings in 2004. In 2000 (the last election in which it voted Republican), Bush won it 52-39% against Al Gore, with Ralph Nader taking 7% of the vote. Bush carried it by 13% that year. Four years later, John Kerry won it 53-45%, carrying it by 8%. Kerry clearly picked up Nader's vote, but he also won over a substantial number of Bush voters. I'm assuming that these were "country club" Republicans of the sort that were already beginning to trend against the Party, even before the arrival of Trump on the scene. Teton County is known for its wealthy, well-educated population, so this would make sense.
It may also have been because of migration of wealthy liberals as Jackson Hole became a well-known, fashionable ski resort. Between 2000 and 2004, Bush only lost 300 votes, third parties lost 700, but Kerry gained almost 2,000 votes for the Democrats.
Similarly, Trump got roughly the same number of votes as Dole in 1996, but Hillary had over 3,000 more than her husband; from 1990 to 2000 the county saw a huge population increase of 63.4%.
