In Crafty, I extend when I check the opponent, rather than extending to escape check, since that made things a bit more accurate. Now, in the normal search, if I check the opponent, I can _never_ get directly to Quiesce() since the search is extended, which gives a bit more accuracy than dropping into q-search when in check. This one-reply-to-check extension extends on the escape, which means two consecutive plies are extended, once to give the check, and once to escape the check if there is only one legal move.
Here's the two results:
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Crafty-23.4-1 2674 3 3 30000 65% 2559 22%
Crafty-23.4R05 2649 4 4 21900 62% 2558 22%
I decided to run this since this idea is an obvious feature of singular extensions, in that if you only have one legal move to try in a position, it would always be extended. The difference is that it would be possible in some positions to only have one move but not be in check, particularly in endgames. I'd think, logically, that if SE was really going to help, this would be a place where it might make sense to extend. But apparently not. I have one more run with longer time controls, for verification.