We show that given $α\in (0, 1)$ there is a constant $c=c(α) > 0$ such that any planar $(α, 2α)$-Furstenberg set has Hausdorff dimension at least $2α+ c$. This improves several previous bounds, in particular extending a result of Katz-Tao and Bourgain. We follow the Katz-Tao approach with suitable changes, along the way clarifying, simplifying and/or quantifying many of the steps.