Second-order information is valuable for many applications but challenging to compute. Many works focus on computing or approximating Hessian diagonals, but even this simplification introduces significant additional costs compared to computing a gradient. In the absence of efficient exact computation schemes for Hessian diagonals, we revisit an early approximation scheme proposed by Becker and LeCun (1989, BL89), which has a cost similar to gradients and appears to have been overlooked by the community. We introduce HesScale, an improvement over BL89, which adds negligible extra computation. On small networks, we find that this improvement is of higher quality than all alternatives, even those with theoretical guarantees, such as unbiasedness, while being much cheaper to compute. We use this insight in reinforcement learning problems where small networks are used and demonstrate HesScale in second-order optimization and scaling the step-size parameter. In our experiments, HesScale optimizes faster than existing methods and improves stability through step-size scaling. These findings pave the way for scaling second-order methods in larger models in the future.