This paper presents a new methodology to develop “thiol-epoxy” shape-memory polymers (SMPs) with enhanced mechanical properties in a simple and efficient manner via “click” chemistry by using thermal latent initiators. The shape-memory response (SMR), defined by the mechanical capabilities of the SMP (high ultimate strength and strain), the shape-fixation and the recovery of the original shape (shape-recovery), was analyzed on thiol-epoxy systems by varying the network structure and programming temperature. The glass transition temperature (Tg) and crosslinking density were modified using 3- or 4- functional thiol curing agents and different amounts of a rigid triglycidyl isocyanurate compound. The relationship between the thermo-mechanical properties, network structure and the SMR was evidenced by means of qualitative and quantitative analysis. The influence of the programming temperature (Tprog) on the SMR was also analyzed in detail. The results demonstrate the possibility of tailoring SMPs with enhanced mechanical capabilities and excellent SMR, and intend to provide a better insight into the relationship between the network structure properties, programming temperature and the SMR of unconstrained (stress-free) systems; thus, making it easier to decide between different SMP and to define the operative parameters in the useful life.