Opinion polls show Hyde-Smith, a former Democrat who has been endorsed by Trump, as the front-runner in the race. Democrat Mike Espy, a former congressman and agriculture secretary in the Clinton administration, is in second place and drawing enough support to possibly make it into a runoff.
Mississippi State Senator Chris McDaniel, a conservative Republican firebrand who tried to unseat Cochran in 2014, is third in the polls, with Democrat Tobey Bartee a distant fourth.
Although Mississippi has not sent a Democrat to the U.S. Senate since 1982, some observers say the increased attention and advertising prompted by a runoff could result in unusually high voter turnout that makes it more of a horse race.
Turnout was cited as a key factor in Democrat Doug Jones upset victory in the U.S. Senate special election in neighboring Alabama last year.