I’ll hazard a response that the 1967 campaign is the one to rival this one, as far as being in it until the very end. 1967 was the year I was born, so unfortunately (or fortunately, perhaps?) this is not personal recollection.
A three-game sweep in the last series of the year against the very bad Washington Senators was what the sox needed to advance. Instead, the Senators swept the Sox, who finished with a record of 89-73. Like the 2003 team, the 1967 he team was apparently a bad home team, as they were 49-33 at home and 40-40 while playing on the road.
The 1977 campaign is the heartbreak "almost made it" year I do remember. But of course we were really pretty much out of it the whole month of September, when the Royals kept playing insanely great ball and we did not.