I am Orthodox and I wear a wig at work, and at formal functions, but usually I just wear a hat or a scarf.
I also have never been arraigned for welfare fraud or any other crime, but I did have my driver’s license photo taken wearing a head scarf.
OK, I did a google search - with pictures - on Orthodox Jewish women wigs.
I found nothing that covered the face.
Now if it’s a hat, then isn’t it standard procedure to remove all hats for mug shots?