On the night that Thomas picks up June and tells her that something's gone wrong at the hospital and that her brother is dead, he shows up with rifle grease on his forehead. This may seem like a minor detail at first, but the rifle grease comes to symbolize much more—it is the sign that Thomas is tainted with Metias's death. His blood is on his hands:
These black marks look like rifle grease. Almost like the streak of grease that was on Thomas's forehead when I first saw him that night. (2.8.36)
The rifle grease represents Thomas's treachery and lies. He's sullied as a person by the way that he's lied to June about her brother's death, and the way that he simply follows orders without thinking of the human consequences. And insofar as Thomas is just a yes man for the Republic, the grease symbolizes the systemic oppression—of people and of truth—that haunts everyone in this book.