In the future, urban regions will encounter a massive number of capacity-hungry devices. Relying solely on terrestrial networks for serving all UEs will be a cost-ineffective approach. Consequently, with the anticipated supply and demand mismatch, several UEs will be unsupported. To offer service to the left-out UEs, we employ an energy-efficient and cost-effective beyond-cell communications approach, which uses reconfigurable intelligent surfaces (RIS) on a high-altitude platform station (HAPS). Particularly, unsupported UEs will be connected to a dedicated control station (CS) through RIS-mounted HAPS. A novel resource-efficient optimization problem is formulated that maximizes the number of connected UEs, while minimizing the total power consumed by the CS and RIS. Since the resulting problem is a mixed-integer nonlinear program (MINLP), a low-complexity two-stage algorithm is developed. Numerical results demonstrate that the proposed algorithm outperforms the benchmark approach in terms of the percentage of connected UEs and the resource-efficiency (RE). Also, the results show that the number of connected UEs is more sensitive to transmit power at the CS than the HAPS size.