If you haven't already, make the switch over to Back-Button Focusing. Although it may take some time to get used to, BB Focusing is far superior to auto focusing by half-pressing the shutter.
If you want to continuously track a moving subject then you simply hold in the Back Button, allowing you to take the shot at any moment without having to re-adjust your focus due to half pressing the shutter. BB Focusing is also perfect for landscape photography, you can lock your focus on your subject (for eg. a log bottom right of the frame), then recompose and take the shot without the camera trying to re-adjust the focus.
Since we're limited to five sentences this is difficult to explain in detail, so I recommend watching the following YouTube video: