I don't think it really hurts to store a move for fail-low nodes. You are not sure it is the best, because its score is only an upper bound, so it might be arbitrarily worse. But it is always better to start with that move, than with one for which you already know for sure that it is very bad, because it has an even lower upper bound.
Note that the first time you encounter a node is usually at low depth, (if not, you should do IID), and that low-depth searches are often not bad at getting the true value. In particular in QS, if you do a capture that is bad, there is often only one way for the opponent to refute it. So he will find that one way, and thus get the true score.
If the moves you normally try first (e.g. MMV/LVA HxL captures of a piece that turns out to be defended, or good captures according to SEE with a soft-pinned or overloaded piece, or even more common, because a threat already exists to one of your other pieces,) are punished badly, it would be quite useless to start trying these moves again. If the node keeps failing low, the order did not matter anyway, but just on the chance that it might fail high at the new depth, you want to at least start with a move that has some chance.
It might be an idea to use the hash move in a fail-low entry to inform the search that all moves before the given one in the normal move ordering are crap, and should be tried last.
