Just because the skies are white doesn't necessarily mean that they should be kept out of the frame.
While cloud-covered white skies are sometimes welcomed, especially for the broad even light they provide, they are not usually my favorite for image backgrounds and I often avoid the inclusion of white skies in image backgrounds.
However, they can be used to create a sometimes-desirable pure white high key background.
Getting this background is not difficult.
Simply find a good subject and align it with the white sky.
Note that your camera's meter will want to make a white sky grey (especially if the subject is a white goat) so some positive exposure compensation (or a manual exposure) will likely be needed for such images.
On this day, my daughter and I were chasing mountain goats high in the Rockies and as you have already figured out, the skies were white.
The thick cloud cover meant that we could photograph the goats from any angle offered to us without concern for shadows but any sky in the photo was going to be white.
Getting into a position that allowed the entire background to be sky and allowing that background to become pure white created a nice portrait.
The versatile and optically-impressive Canon EF 100-400mm f/4.5-5.6L IS II USM Lens
was a great lens to have for this trip.
It was the only lens I used for photographing the goats.