The 33-year-old Hart, who's made seven starts at first base and three in right field during his first season in Pittsburgh, is out because of a left shoulder impingement.
Volstad, a right-hander, allowed two or fewer runs in six of his past seven starts for Triple-A Indianapolis. The 28-year-old was 5-2 with a 3.43 ERA in 76 innings over 14 appearances (12 starts). He will pitch out of the bullpen for Pittsburgh.
Signed as a minor-league free agent in January, Volstad has not pitched in the majors since June 2013. A first-round draft choice by Florida 10 years ago, Volstad has a 4.62 ERA in 130 games (123 starts).
The Pirates, who have the second-best record in the National League, will go with a four-man bench for the time being.