  Let $\mathcal{X}$ be a Riemann surface of genus $g>0$ defined over a number
field $K$ which is a degree $d$-covering of $\mathbb{P}^1_K$. In this paper we
show the existence of infinitely many linearly disjoint degree $d$-extensions
$L/K$ over which the Jacobian of $\mathcal{X}$ gains rank. In the case where 0,
1 and $\infty$ are the only branch points, and there is an automorphism
$\sigma$ of $\mathcal{X}$ which cyclically permutes these branch points, we
obtain the same result for the Jacobian of $\mathcal{X}/\sigma$. In particular
if $\mathcal{X}$ is the Klein quartic, then the construction provides an
elliptic curve which gains rank over infinitely many degree $7$-extensions of
$\mathbb{Q}$. As an application, we show the existence of infinitely many
elliptic curves that gain rank over infinitely many cyclic cubic extensions of
$\mathbb{Q}$.
