  We consider Steklov eigenvalues of nearly hyperspherical domains in
$\mathbb{R}^{d + 1}$ with $d\ge 3$. In previous work, treating such domains as
perturbations of the ball, we proved that the Steklov eigenvalues are analytic
functions of the domain perturbation parameter. Here, we compute the
first-order term of the asymptotic expansion and show that the first-order
perturbations are eigenvalues of a Hermitian matrix, whose entries can be
written explicitly in terms of the Pochhammer's and Wigner $3j$-symbols. We
analyse the asymptotic expansion and show the following isoperimetric results
among domains with fixed volume: (1) for an infinite subset of Steklov
eigenvalues, the ball is not optimal, and (2) for a different infinite subset
of Steklov eigenvalues, the ball is a stationary point.
