  We provide the phase diagram of a 2-parameter spin-1 chain that has a
symmetry-protected topological (SPT) Haldane phase using computational
algorithms along with tensor-network tools. We improve previous results,
showing the existence of a new phase and new triple points. New striking
features are the triple end of the Haldane phase and the complexity of phases
bordering the Haldane phase in proximity allowing moving to nearby non-SPT
phases via small perturbations. These characteristics make the system, which
appears in Rydberg excitons, e.g. in Cu$_2$O, a prime candidate for
applications.
