The CGI deer ...

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That was a real deer, it says so in the trivia:
The "deer scene" was filmed with Becca, the White-Tailed Deer at the Western North Carolina Nature Center. Becca's trainer/caretaker is award-winning singer/songwriter, Ben Wilson.

Oh man. That was brutal. Took me right out of the movie. I mean, how hard is it for a major film production to get some footage of a real goddamn deer, and then splice that into the scene through effects. As it was, the scene was embarrassingly terrible-looking.

Maybe there can be a Three Billboards Outside Ebbing, Missouri: Special Edition where they fix that deer.

Jokes aside, it did look out of place. It's a good scene (the deer reminded her of her daughter) but it did seem like that deer arrived via 1996 special effects.