Insert a link within header image

I am trying to reduce the amount of space used by this (excellent) theme.

I have removed the title (and most of the white space it left) and also replaced the default menu with a ‘blank’ one)

Is there a way that I can insert a ‘home / logo’ within the main header. so that it sits ‘over’ the image?

Perhaps having a ‘home’ button by the social media links would solve my problem?

Any help welcome.