Just because we lacked knowledge of how a wound becomes infected does not mean we couldn't figure out bad things happen when you let a wound get dirty. While it focuses on Early Medieval England, there's nothing there that wouldn't be available to pretty much any other society even into the BCs, so here is u/BRIStoneman's post on the medical textbook Bald's Leechbook and what it advises for infections.

As always, more input is always welcome! Pre-modern medicine is not nearly as bad as people think it is, and additional posts on this matter would be a great help.