The first doubleheader was in Philadelphia against the As. The rest at Yankee Stadium: two against the White Sox, then three against the Indians. Yanks were 9-3. The Yankees played 26 doubleheaders in 1938.
I thought that was a lot of doubleheaders so close together that late in baseball history. So I did a search.
Most consecutive doubleheaders: 1928 Boston Braves Sept. 4-15: 9:
Most consecutive doubleheaders between the same teams: Philadelphia v. Washington five Aug. 5-10, 1901.
Most doubleheaders in one season: Chicago White Sox 44 in 1943, in the middle of World War II when travel was limited. In the 1943 World Series the Yankees beat the Cardinals 4-1; the first three games were played in New York; the remaining four would have been played in St. Louis to eliminate one trip had the format been the traditional 2-3-2.
Friday Aug. 23 doubleheader Yankee Stadium v. Detroit (W-T; I attended)
Saturday Aug. 24 single game Yankee Stadium v. Detroit W
Sunday Aug. 25 doubleheader Yankee Stadium v. Detroit (W (winning pitcher: outfielder Rocky Colavito) -W)
Monday Aug. 26 doubleheader Yankee Stadium v. California (W-L)
Tuesday Aug. 27 doubleheader Yankee Stadium v. California (W-L)
Then from Aug. 28 through Sept. 13 off days on Sept. 5, 6, 9; doubleheaders:
Aug. 30 (W-L)
Sept. 2 (W-L)
Sept. 7 (W-W)
Sept. 10 (W-W)
Sept. 13 (W-W).
In 1961 the American League expanded and changed the schedule from 154 to 162 games. The National League did the same in 1962.