In the Header.php file, is there a way to swap between one layout and another?

Depending on the theme, in most cases you can simply edit header.php in your child theme and add a conditional:

if(is_home()) {
// code for the header displayed on home & paged
} else {
// code for the header on all other URLs
}

is_home() will return true if you are on what is referred to as the “blog posts index” – i.e. you’re displaying a list of blog posts on the front page – as well as each additional page you reach through the “next” link.