Smith was traded to the Giants on Monday, Jeff Passan of Yahoo Sports reports.
The 27-year-old put together a 3.68 ERA with 22 strikeouts in 22 innings for the Brewers. He'll now join fellow left-handers Javier Lopez and Matt Reynolds in the Giants bullpen. In return, the Brewers will receive right-handed pitching prospect Phil Bickford and catcher Andrew Susac.