All of the following have hunting scenes to various degrees.
Thriller/ Action : No Country for Old Men
A Vietnam Vet goes hunting and finds the aftermath of a shootout near the Mexican border along with a large sum of cash. Having taken it, he is pursued by the ruthless and relentless Anton Chugur. The aging police officer looking into the case follows a brutal trail, eventually concluding that it is 'No Country for Old Men'
Family/Humour: Hunt for the Wilderpeople
A Taika Waititi film depicting the adventures of an adopted boy and his reluctant foster father who land up in the NZ bush, a misunderstanding causes them to go on the run from the NZ authorities. Good fun and kid friendly.
Drama/War: The Deer Hunter
Probably one of the best American films ever made, can be slow but very hard hitting if you are in the target audience. Three friends who are steelworkers join the Army to serve in Vietnam, coming from a Russian community in Pennsylvania. Career defining performance from De Niro and Walken whose friendship is reflected by their shared passion for deer hunting. Their mantra of 'One Shot' being an important theme. Very reflective as it was one of the first movies to depict the Vietnam War after it ended. An emotional watch.