The ever-increasing number of users and new services in urban regions can lead terrestrial base stations (BSs) to become overloaded and, consequently, some users to go unserved. Compounding this, users in urban areas can face severe shadowing and blockages, which means that some users do not receive a desired quality of service (QoS). Motivated by the energy and cost benefits of reconfigurable intelligent surfaces (RIS) and the advantages of high altitude platform stations (HAPS), including their wide footprint and strong line-of-sight (LoS) links, we propose a solution to service the stranded users using the RISaided HAPS. More specifically, we propose to service the stranded users by a dedicated control station (CS) via a HAPS equipped with RIS (HAPS-RIS). Through this approach, users are not restricted from being serviced by the cell they belong to; hence, we refer to this approach as beyond-cell communication. As we demonstrate in this paper, beyond-cell communication works in tandem with legacy terrestrial networks to support uncovered or unserved users. Optimal transmit power and RIS unit assignment strategies for the users based on different network objectives are introduced. Numerical results demonstrate the benefits of the proposed beyond-cell communication approach. Moreover, the results provide insights into the different optimization objectives and their interplay with minimum quality-of-service (QoS) and network resources, such as transmit power and the number of reflectors.