In this paper, we derive a set of first-order Pontryagin optimality conditions for a risk-averse stochastic optimal control problem subject to final time inequality constraints, and whose cost is a general finite coherent risk measure. Unlike previous contributions in the literature, our analysis holds for classical stochastic differential equations driven by standard Brownian motions. Moreover, it presents the advantages of neither involving second-order adjoint equations, nor leading to the so-called weak version of the PMP, in which the maximization condition with respect to the control variable is replaced by the stationarity of the Hamiltonian.