It's hard to miss any of the clues Old Smokey has put down. The only clue about engine damage (that I can think of, that is), after the engine has flamed out, would be oil streaks on engine nacelle or wing but it's far from reliable indication.
So would I try to restart engine if it flames out quietly or violently but unnoticed (just a hypothetical case)? Absolutely yes - using inflight restart procedure and applying caution. If vibrations start, if EGT rises fast nad high, if oil press stays low then it's start abort and single engine ops time.