How to add markup to the wp-custom-header div

I have a custom header video on my WordPress site and I have an overlay (an svg I repeat in the background of a div) that I want to live on top of the video. I accomplished this like so

?php the_custom_header_markup(); ?
div class="header__overlay"/div

However this creates a problem. Because the video and the generated pause button are both in the generated wp-custom-header div, I cannot get my overlay between them. ie For my overlay to be z-indexed over my video, it is also over top of my button.

Is there a way for me to filter the output of The_custom_header_markup to include additional markup. That functions typical markup is:

div id="wp-custom-header" class="wp-custom-header"
video id="wp-custom-header-video"/video
button type="button" id="wp-custom-header-video-button" class="wp-custom-header-video-button ..."Pause/button
/div

What I want is

div id="wp-custom-header" class="wp-custom-header"
video id="wp-custom-header-video"/video
div class="header__overlay"/div
button type="button" id="wp-custom-header-video-button" class="wp-custom-header-video-button ..."Pause/button
/div

Thanks In advance! JoesLost