I have 3 sets of damage counters including the ones from the original starter set and combined them.
I still like to use the original ships logs rather than clean off ones like The Admiral and Dobbs do, but it's just an aesthetic thing because I believe a laminated card is a more convenient option. Note I'm still pretty much a novice having only played 15 to 20 games and all but 1 of them solo, maximum of 8 ships at any one time.