To achieve a silver rating on an animal, you will need to score a certain number of points depending on the species. An animal's difficulty rating (typically increasing with the size of the animal) is a large portion of the score.
In addition to hunting more difficult animals, there are some other factors that determine your score:
Using the appropriate weapon class for the specific animal, taking two or fewer shots to down the animal, hitting a vital organ, and not destroying any trophy organs.
This chart shows the minimum score required for each species:
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