When you reach the point where one shot will not kill the animal you are hunting and you are unable to upgrade your gun to make it more powerful you will need to move quickly to reload your gun to have any chance of finishing the animal off.
Upgrade To get enough cash to max out any gun will take you a long time, it is better to let the mission system guide you towards your next upgrade. A list of suggested upgrades will appear before you go on a hunt, work towards getting these.
Infrared There will be certain missions that you will need to have the infrared power activated as they will involve shooting an animal in the vital organs. The heart sits in the middle, and the lungs are either side of the heart. Before a mission starts make a point of double checking the objective.
Ducks The easiest way of shooting down the ducks in the contract hunting work is to establish the general direction the bird is flying and move your gun sights a little ahead of it's flight path and then fire when it flies though your crosshairs. Don't worry if you don't hit the duck dead center, all that matters is that you strike within the general hit window.
Energy Meter If you do not have the patience to wait for the energy meter to refil you can use some of the gold bars that you earn to top it up straight away. Use this precious resource wisely as there may be a point in the game you need it ugently.
Later Levels When you reach the later levels and you find that a single shot does not kill an animal you need to either upgrade your gun so it is more powerful or position yourself before the shot so that everything is lined up and when you do fire it is a critical shot. It also helps if you can reload fast just in case you need to take another shot to finish the job.
Secondary Weapon Some animals like wolves will fight back if they are not hit critically so make sure you have your shotgun ready just in case you didn't kill your prey.
Animal Call If you are struggling with a mission you can use an item called an 'Animal Call'. When you activate this special item you will restart the mission retaining the animal kills you racked up in he earlier attempt. Try to use these wisely as you only get a few for free.
Preview In the early stages of the game the hunting scene will only have animals you need to kill but as you progress you will come upon a mixture of animals wandering around the landscape. It is when you reach these later stages that it is important that you pay attention when you observe the animal preview before you begin as it shows you the appearance of your prey.