We consider a continuous-time bilinear control system with Metzler matrices. The transition matrix of such a system is entrywise nonnegative, and the positive orthant is an invariant set of the dynamics. Motivated by the stability analysis of positive linear switched systems (PLSSs), we define a control as optimal if, for a fixed final time, it maximizes the spectral radius of the transition matrix. A recent paper  developed a first-order necessary condition for optimality in the form of a maximum principle (MP). In this paper, we derive a stronger, second-order necessary condition for optimality for both singular and bang-bang controls. Our approach is based on combining results on the second-order derivative of the spectral radius of a nonnegative matrix with the generalized Legendre-Clebsch condition and the Agrachev-Gamkrelidze second-order variation.