This paper investigates the convergence performance of second-order needle variation methods for nonlinear control-affine systems. Control solutions have a closed-form expression that is derived from the first-and second-order mode insertion gradients of the objective and are proven to exhibit superlinear convergence near equilibrium. Compared to first-order needle variations, the proposed synthesis scheme exhibits superior convergence at smaller computational cost than alternative nonlinear feedback controllers. Simulation results on the differential drive model verify the analysis and show that second-order needle variations outperform first-order variational methods and iLQR near the optimizer. Last, even when implemented in a closed-loop, receding horizon setting, the proposed algorithm demonstrates superior convergence against the iterative linear quadratic Gaussian (iLQG) controller.