The Illyrians had starship technology a hundred years before SNW, so I'm sure that by the time of the latter, they were in several star systems.
Guess they must have gotten over what Archer did to them in "Damage"!
From what the writers said on Twitter, and from the screen images depicted when Una asked for info on Illyrian modifications, apparently the implicit thinking was that the "Damage" Illyrians were themselves a heavily modified offshoot of a more humanlike root species, explaining why they look so different from Una. Also, the Illyrian crew in "Damage" said they were new to that region of space and three years from their home system -- which, again, could well have been a colony. So what Archer did to that one far-flung ship probably had little effect on the civilization as a whole.
Really, though, the species in "Damage" was never called Illyrian onscreen; the name comes only from the script. So strictly speaking, we don't have any canonical proof that the species in "Damage" even were Illyrians.