Unless something changes, it looks like the Sox will face RHP in their next 9 consecutive games, and in 13 of their next 14 (Only LHP they're scheduled to face is Moore in the 2nd TB series).
Sox vs RHP: .288/.362/.484/.847
Sox vs LHP: .251/.333/.404/.737
This bodes well for the Sox, especially since they've got a pretty tough schedule in June, at least the next 14 games (3 @ TB, 4 @ BAL, 3 v TB, 4 @ DET).