Baseball is fairly slow to adjust strategies, but here's an obvious one.
With the new extra-inning rules, a runner starts on second base.
Therefore, I think if it's a tie game in the 8th+, the best strikeout relief pitchers should be saved for the tenth.
Maximize the chance of stranding the runner at 2nd, and let your 2nd tier relievers handle the 8th/9th.
For example, the Pirates just blew their closer in the 9th inning of a tied game against the bottom of the Reds order, leaving a 2nd tier reliever to handle the 10th inning, with the top of the order and a runner on 2nd. Needless to say, it didn't end well.