We study off-diagonal Ramsey numbers
$r(H, K_n^{(k)})$ of
$k$-uniform hypergraphs, where
$H$ is a fixed linear
$k$-uniform hypergraph and
$K_n^{(k)}$ is complete on
$n$ vertices. Recently, Conlon, Fox, Gunby, He, Mubayi, Suk, and Verstraëte disproved the folklore conjecture that
$r(H, K_n^{(3)})$ always grows polynomially in
$n$. In this paper, we show that much larger growth rates are possible in higher uniformity. In uniformity
$k\ge 4$, we prove that for any constant
$C\gt 0$, there exists a linear
$k$-uniform hypergraph
$H$ for which
\begin{equation*} r(H,K_n^{(k)}) \geq {\textrm {twr}}_{k-2}(2^{(\log n)^C}). \end{equation*}