what distance do you guys shoot animals so they do not run away like the flash?

I am level 31, carrying 3 rifles with me. 1: Virant .22LR, ranger .243, and Arzyna .300 mag tactical. I have no problem shooting anything down, but when I hunt, I like to get within 20-100m and take my shot. The question I have is, at what distance do you shoot the following animals: Coyote, black-tailed deer, and white-tailed deer? I ask this question, because by the time i do a 2nd shot every single time the animals are running like flash and never ever have time to take good 2nd shots, so what i resort to is shooting the best trophy with the .243 and switch rigth to the .300 mag and wait till that burst of speed goes away and shoot them in the ass, but they still go down. its the initial burst of speed like they are the flash that messes me up. Is there a certain distance I have to shoot them so they get spooked and just run away normally?