We show that, for every $k \ge 2$, every $k$-uniform hypergaph of degree $Δ$ and girth at least $5$ is efficiently $(1+o(1) )(k-1) (Δ/ \ln Δ)^{ 1/(k-1) } $-list colorable. As an application (and to the best of our knowledge) we obtain the currently best algorithm for list-coloring random hypergraphs of bounded average degree.