Is there a plugin or mechanism to insert html for a navigation bar into the header?

I have a navigation bar for my normal (non-WP) Website in my header.php, see the top of http://blog.atomicinc.com.

I did it by inserting some HTML (div > ul > li) directly into header.php, along with some styles into style.css, but this means I have a custom theme, theme lock, etc.

I would like to extract this into something managed at the WP or plugin level, so I can go back to standard themes, upgrade, etc.

I use https://wordpress.org/plugins/insert-headers-and-footers/ to insert scripts into head; is there anything similar that will allow me to do it html inside body in header.php?