  We study the derivative of the standard $p$-adic $L$-function associated with
a $P$-ordinary Siegel modular form (for $P$ a parabolic subgroup of
$\mathrm{GL}(n)$) when it presents a semi-stable trivial zero. This implies
part of Greenberg's conjecture on the order and leading coefficient of $p$-adic
$L$-functions at such trivial zero. We use the method of Greenberg-Stevens. For
the construction of the improved $p$-adic $L$-function we develop Hida theory
for non-cuspidal Siegel modular forms.
