Answer by taking a quote from the following article:

Benny was born in Chicago, Illinois, and grew up in nearby Waukegan, Illinois. He was the son of Meyer Kubelsky and Emma Sachs Kubelsky. His parents were Jewish. Meyer was a saloon owner and later a haberdasher who had emigrated to America from Poland.

Benny had been a minor vaudeville performer before becoming a national figure with The Jack Benny Program, a weekly radio show that ran from 1932 to 1948 on NBC and from 1949 to 1955 on CBS. It was among the most highly rated programs during its run.  Benny's long radio career began on April 6, 1932, when the NBC Commercial Program Department auditioned him for the N.W. Ayer agency and their client, Canada Dry, after which Bertha Brainard, head of the division, said, "We think Mr. Benny is excellent for radio and, while the audition was unassisted as far as orchestra was concerned, we believe he would make a great bet for an air program." Recalling the experience in 1956, Benny said Ed Sullivan had invited him to guest on his program (1932), and "the agency for Canada Dry ginger ale heard me and offered me a job."  With Canada Dry ginger ale as a sponsor, Benny came to radio on The Canada Dry Program, on May 2, 1932, on the NBC Blue Network and continuing for six months until October 26, moving to CBS on October 30. With Ted Weems leading the band, Benny stayed on CBS until January 26, 1933.  Arriving at NBC on March 17, Benny did The Chevrolet Program until April 1, 1934. He continued with sponsor General Tire through the end of the season. In October, 1934, General Foods, the makers of Jell-O and Grape-Nuts, became the sponsor strongly identified with Benny for 10 years. American Tobacco's Lucky Strike was his longest-lasting radio sponsor, from October 1944 through to the end of his original radio series.  The show switched networks to CBS on January 2, 1949, as part of CBS president William S. Paley's notorious "raid" of NBC talent in 1948-49. It stayed there for the remainder of its radio run, ending on May 22, 1955. CBS aired repeat episodes from 1956 to 1958 as The Best of Benny.

What was something interesting Jack Benny did in this period?
becoming a national figure with The Jack Benny Program,