Hamilton was injured while trying to make a diving catch in Tuesday's 3-1, 13-inning loss to Kansas City. Hamilton, who finished second in National League Rookie of the Year balloting last year, is batting .226 with a major league-leading 54 stolen bases.
Cincinnati also selected the contract of right-hander Sam LeCure, recalled outfielder Brennan Boesch from Triple-A Louisville, optioned right-hander Dylan Axelrod to Louisville and transferred righty Nate Adcock to the 60-day DL.
LeCure has spent the entire season with Louisville after being a surprise cut during spring training. He spent most of the previous five seasons with the Reds.
Boesch opened the season with the Reds and played in 32 games before he was optioned to Triple-A on June 16.