  We consider the zero-average Gaussian free field on a certain class of finite
$d$-regular graphs for fixed $d\ge 3$. This class includes $d$-regular
expanders of large girth and typical realisations of random $d$-regular graphs.
We show that the level set of the zero-average Gaussian free field above level
$h$ has a giant component in the whole supercritical phase, that is for all
$h<h_\star$, with probability tending to one as the size of the graphs tends to
infinity. In addition, we show that this component is unique. This
significantly improves the result of [AC20b], where it was shown that a linear
fraction of vertices is in mesoscopic components if $h<h_\star$.
