The straight segment of a racetrack leading to the finish line.
2.
Hence: (fig.) The last or concluding part of any extended activity that has a definite conclusion; the final phase of any activity; as, the election was neck-and-neck in the homestretch.
... Cree, shot the Yankees to pieces. For nearly six weeks, however, it was a battle royal between New York and St. Louis to escape the last hole, but in the middle of August the Yankees again established their superiority, retaining seventh place until after the middle of September. In the homestretch the new blood given Stovall enabled him to pull his men out of the last notch just before the schedule ran out. This feat was soon forgotten in the defeat of the Browns by the Cardinals in their post-prandial series for the championship of ... — Spalding's Official Baseball Guide - 1913 • John B. Foster