In many mean-field glassy systems, the low-temperature Gibbs measure is dominated by exponentially many metastable states. We analyze the evolution of the metastable states as temperature changes adiabatically in the solvable case of the spherical s + p-spin glass model, extending the work of Barrat et al (1997 J. Phys. A: Math. Gen. 30 5593). We confirm the presence of level crossings, bifurcations, and temperature chaos. For the states that are at equilibrium close to the so-called dynamical temperature Td, we find, however, that the following state method (and the dynamical solution of the model as well) is intrinsically limited by the vanishing of solutions with non-zero overlap at low temperature.