From what I've seen in regards to this gene mod, this "+10 aim" applies rather often based on what circumstances take place. What I've seen:
Close and Personal, Double Tap, Bullet Storm-(If the first shot misses, the bonus is applied to the second shot...should you choose to take it). HOWEVER, if a previous shot was made (overwatch or the turn before) and missed, the effect will be applied to the first and will apply to the second IF the first shot misses.
Reaction Overwatches (that miss) will also confer the bonus FOR THE NEXT SHOT (this means for either the next turn OR if the character has the Sentinel perk with two shots [assuming that was the first shot...if it was the second, then the next turn's shot has the bonus])
Some additional obversations:
I have noticed that using a rocket launcher for heavies (regardless of whether it hits or not), counts as a miss and therefore applies the bonus for those characters.
Opportunist with this mod make it rather hard for them to NOT miss. Perfect for those snipers.
If your character has this bonus and you choose to suppress targets, you can lose the bonus if it hits. Same with Flush (Don't know why you'd want to do that though xD since it generally is pretty easy to hit).
This is very useful for Disabling shot Snipers if a soldier gets "mind-controlled" since they have a better chance to prevent their ally from hurting their squadmates.