How to change the order of the menu items or add a custom menu item with hook

Hi, I'm using the sliding panel on a site, so the toggle element is the last in the primary menu.
I need to achieve this look:
https://dashnexagency.com/contacts/
where before the toggle there's a button
So I tried to float it right - it locates itself after the toggle.
I've tried to add button to the end of the menu through the wp_nav_menu_items filter - no luck, it returns a blank menu.
Is there a workaround in this situation?
The feature to locate the item after the toggle would be good.
Thanks for the help.