Winds of the Wasteland (1936) on Netflix

Plot summary

In 1861 the telegraph brings the Pony Express to an end, and its riders John Blair and Larry Adams look for work with horses. In Buchanan City they ask the stage line owner Cal Drake to sell them a coach, and he offers them his franchise out to Crescent City instead. The town, they find, is almost deserted, with Mayor Rocky O'Brien and Dr. William Forsythe its only residents. O'Brien says a race can earn them a $25,000 government contract to carry the region's mail, and Blair goes to Buchanan City to enter. Barbara Forsythe, the doctor's daughter, arrives from the east on Blair's stage to join her father, who has not told her the town is nearly empty.