Thank you so much for using our theme and thanks for your questions too. If you need to add the overlay on the inner pages top image, it needs a customization of theme. So, we recommend you to make the overlay on that header image through photoshop and add that image in place of header image. Go to Appearance > Customize > Header Options > Header Image. Hope, this will help you to make as your need.
If you have any confusion, let us know.
Many thanks for your kind reply.
I understand and appreciate your suggestion.
In that case, does this mean that the homepage header image (the large one) will also get the photoshoped version?
Should I then rather completely disable the CSS overlay (code above at my initial message) at homepage?
Header image will be the slider background image and inner image as well. If you make the image from photoshop, and place it on header image, it will appeared as it is on the inner pages. If you need any help regarding it, let us know. We will provide you the best help ever.