For optimal control problems with set-valued control constraints and pure state constraints we propose new second-order necessary optimality conditions. In addition to the usual second-order derivative of the Hamiltonian, these conditions contain extra terms involving second-order tangents to the set of feasible trajectory-control pairs at the extremal process under consideration. The second-order necessary optimality conditions of the present work are obtained by using a variational approach. In particular, we present a new second-order variational equation. This approach allows us to make direct proofs as opposed to the classical way of obtaining second-order necessary conditions by using an abstract infinite dimensional optimization problem. No convexity assumptions on the constraints are imposed and optimal controls are required to be merely measurable.