Sorry for the wait on this.
I went ahead and changed the hook to generate_after_mobile_header_menu_button
as you’re using the mobile header.
I also gave it a wrapper:
<div class="mobile-header-custom-image-container">
your image HTML here
</div>
Now you should be able to add this CSS:
.mobile-header-custom-image-container {
order: 4;
position: relative;
width: 100px;
height: 60px;
overflow: visible;
}
.mobile-header-custom-image-container img {
position: absolute;
top: 0;
right: 0;
z-index: 10;
}