We propose a new method for linear second-order cone programs. It is based on the sequential quadratic programming framework for nonlinear programming. In contrast to interior point methods, it can capitalize on the warm-start capabilities of active-set quadratic programming subproblem solvers and achieve a local quadratic rate of convergence. In order to overcome the non-differentiability or singularity observed in nonlinear formulations of the conic constraints, the subproblems approximate the cones with polyhedral outer approximations that are refined throughout the iterations. For nondegenerate instances, the algorithm implicitly identifies the set of cones for which the optimal solution lies at their extreme points. As a consequence, the final steps are identical to regular sequential quadratic programming steps for a differentiable nonlinear optimization problem, yielding local quadratic convergence. We prove global and local convergence guarantees of the method and present numerical experiments that confirm that the method can take advantage of good starting points and can achieve higher accuracy compared to a state-of-the-art interior point solver.