This will depend upon what aftermath you wish for your character. Physically I mean.
There are many, many ways to produce a "knock-out" hit with your bullet, that will cause little damage - but look huge/deadly (like a head graze) to your hit man. And, there are many that can look like little damage (gut shot - which produces blood yes, but does not always cause immediate passing out - but long term suffering) but be deadly or have long periods needed for rehabilitation.
I do not know this weapon/ammunition and do not know it's abilities to tear flesh or such. Thus, I cannot state what would work best. If you expound on it's capabilities and what happens to the ammo when it hits flesh/bone, I might be able to give you some things that may work. Although, maybe not too. Hard to say for sure.
Lung shots do not cause passing out immediately (usually). They cause gasping and labored breathing and heaving of chest in the victim (whether he passed out or not). It would be a clear signal to your "hit man" that the victim is still alive. Thus, they may go in for another shot. So, think on that one, you may not wish to use it. Although, if your victim can control body response in himself (and does NOT pass out), it is not typically a "life threatening" hit if he can get medical attention swiftly enough that over taxing the other lung does not cause IT to collapse as well.
You do not wish to hit the neck I do not think - but maybe. Try to think if any of this will affect such:
The carotiod and jugular are there - one on each side of the neck. Too easy to hit. Unless your hit man shoots and hits the side of the neck or in fleshy part by/below ear area between ear and shoulder. Anything too close to the front and your victim has those arteries and his windpipe to ruin. All, while may cause passing out, also typically (in most cases) means certain death. So, watch that. Also, the neck houses the cervical vertebra. A shot to neck that goes into flesh could cause damage to those vertebra. Cervical veterbra damaged can (and often does) cause paralysis - for life. And, one does not always pass out from such a hit. Although they WOULD lie still. But, afterward they would likely be unable to move some or all of their body for life. But, there ARE people shot in the side of the neck, it exited the other side and only tore flesh and hit nothing major. It will depends what you wish to do here on that type of shot and how your hitman and victim are physically set apart.
You could have your character be hit anywhere on his body in a flesh part (he turned at the last second, whatever) and have him be smart enough to just lie there - without passing out to give the impression he is dead, passed out, whatever you need to work.
You also can take to heart, that no matter what - some people live and never have bad results from "freak" shots that appear no one could survive.
It will all depend upon what you need your character to do immediately following being hit. If you need this character to be relatively "fit" and able to do his/her job and not be hampered by injury causing lengthy rehabilitaton -then you maybe want to consider "faking" a fatal shot. Or a "freak" shot that missed its mark totally but looked like a kill shot.
If part of your story is that this victim recovers to get revenge and that drives some of it, then you can shoot this person just about anywhere and make it so they can live and survive, but passed out for it. People pass out for all sort of reason. Seeing blood, losing blood causes light headed feeling - thus they swoon, pain causes passing out, falling and hitting head from injury (even minor one) would cause passing out, it depends what comes after for you to determine what will work.
What does happen after? Maybe I could provide more physical examples if I know what you need to occur.
I have worked in ERs. I have seen major and catastrophic injury and people survive such. I have also seen people die from a bruise on their leg with no apparent "Signs" of fatal injury. Each has its own physiology.
Let me know if you have specific questions that I can assist with on the physical stuff. The gun/ammunition - I am lost on that. I cannot tell you what they should do, could do or what they are "said" to do. I can only speak from the aftermath part. The actual physical situational results of such.
Tell me what body part, tell me the size of the hole, tell me if the bullet expands and rips and breaks to pepper internal organs, tell me if it is gonna hit bone and such. I will tell you what I have seen as a result of such. Physically I mean.
Although a doctor (I am not one - just RCP) could provide you substantial info - especially one running an ER.
Good luck!!
Christine