Whatever focus method works for you, go with it.
That being said, let's look at the technical data regarding this shot.
According to the metadata attached to your file; you had the following shooting parameters set on the first image:
Camera: Nikon D800
Lens: f/4.5- f/5.6 80-400mm (I don't know if this is the D or G version of this lens). I own the D version of this lens.
Shot at a focal length of 400mm at ISO 3200 at 1/500th second at f/5.6
Distance to subject 35.5m / 116.ft ft
Nominal DoF - 2.8m / 9.2 ft
At that distance for a decent sized image, your depth of field would be 2.8m / 9.2 ft
That's a pretty decent depth of field to get a sharp eyeball. If you are getting the eye being less than sharp, I would suggest that motion blur is going to be the issue more than a sharp focus. At 1/500th sec you are pretty well at the limit for hand holding and a sharp image.
Let me pull up a wildlife image of mine where I used the focus and recompose method. I was a bit closer and used a slightly shorter focal length, but had a correspondingly much narrower DoF. Instead of a couple of slow moving tigers, I had four wolves that were moving around fairly quickly. They eyes are very sharp. If I had tried your focus method, I'm pretty sure I would have missed the shot as the animals were moving around a lot.
Camera: Nikon D810
Lens: Sigma f/5 - f/6.3 150mm - 500mm
Distance to subject: 12.6m / 41.3 ft
Nominal DoF: 0.5m / 1.6 ft
Shooting data: ISO 3200 with 1/1000th sec shutter speed, f/6.1 aperture and 350mm focal length.