This is an excellent question, since it is the only mention of Maycomb's Mr. Tensaw Jones in the entire novel. The allusion to Jones takes place during the parade of people who pass by on the way to the courthouse to witness the trial of Tom Robinson. Jem is gossiping about the various passersby, primarily for the benefit of Dill, to whom most of them are complete strangers. Apparently, Jones "voted the straight Prohibition ticket," a reference to the Prohibition Party which was still a minor force in national politics of the time. The Prohibition Party adhered strictly to complete temperance and the continued abolishment of alcohol (Prohibition had been repealed two years earlier in 1933), and Jones must have been a teetotaler who voted only for Prohibition Party candidates. The Prohibition Party had its own candidate during the 1928 Presidential election (William F. Varney), but backed President Herbert Hoover (who lost to Franklin Delano Roosevelt) during the 1932 election. With Prohibition repealed, Tensaw Jones must have been one of the most diehard of temperance supporters and probably anti-Roosevelt; the limited power of the Prohibition Party soon waned even further, though it still remains alive today, having
... continued running presidential candidates every four years, but its vote totals have steadily dwindled. It last received more than 100,000 votes for president in 1948, and the 1976 election was the last time the party received more than 10,000 votes for president. In 2008, its presidential nominee received only 643 votes (Wikipedia, Prohibition Party).
Lee probably used Jones and his political preference simply as a way to illustrate the various types of unusual characters who lived in Maycomb: Besides Jones, the author also mentioned the secretive snuff-dipper, Miss Emily Davis; Mr. X Billups, an illiterate man who literally signed his name "X"; and, of course, the "sinful" Dolphus Raymond.