Instead of parenting the weapon to the left hand, parent the left hand to the weapon, that way the left hand should follow the gun around. It works best if the gun has two parts that can't be selected individually. For instance, grip and foregrip or (in this case) magazine/clip. Parent the hand to that, lock the hand pose and it should retain it's position enough that you only need small tweaks to make it look ok. Alternatively what might work if you have seperately parented props. Is leave the right hand parented to the grip, parent the left hand to the secondary prop and parent that prop to the gun. It sounds odd but it can work in some cases.