Second hand low?

One of my weaknesses as defender is that I often play low too quickly in second hand, relying on the maxim "Second hand low."  However, there are cases where this isn't the right play.  One instance is that it is usually right to cover an honor with an honor.  Another instance is when there is a singleton in dummy.

My partner faced this situation the other night.  Declarer was in 3NT and led a small club towards the singleton 9 on the board.  My partner held Jxx, and without much thought played small.  Unfortunately, I held A8x.  Declarer had made an unusual play, holding KQTxxx in his hand, and ultimately scored five tricks in the suit.

The reason it is often right to cover (at least with a low honor like the jack, probably not with the Ace in this case) is to protect against what happened.  Even if I had held Kxx, he needed to rise with the Jack.  Because there was a singleton in dummy, declarer would not be able to subsequently finesse my King.  Similarly, if I held Qxxx, he needed to rise with the Jack in order for us to score two tricks in the suit.   In fact, I can't think of a single holding I could have had (not even AK tight) where ducking the Jack gains us a trick.