Adding a button to the header

Hello
Am using the Enigma-Parallax version and finding my way around WordPress.
Years ago in school I did love coding in C++ and other things, so I do have a basic grounding in code. Still would just like to make this function and get on with work rather than tinkering under the hood, which is taking time, however fun.

In short: the page in question is https://xeniawordart.com/work/
I added the big red ‘appointment’ button as code to the header.php file ; I placed it within li tags in an ul list.
However it appears on a new line. I want it ideally at the end of the social buttons or on the same line as the phone number. On a separate line the header is too big.

I also need to make the header smaller for this page, or remove the header. I think the big header is good for the home page. Subsequent pages will have more content and the big header would be distracting (this issue a separate post perhaps)
Thanks in advance!