Hooking into the HTML header container

I’m trying to hook into the header.php of wordpress theme(s) more specifically, right before the closing header tag </header>. I’d like to do this in a programmatic way so when I switch themes I don’t have to remember to go include the method of a custom hook <?php my_cool_hook(); ?>

Is there a way to accomplish this? I’ve tried looking through the available actions but so far I haven’t found any that would suite my needs.