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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 station (HAPS) systems, including their wide footprint and strong line-of-sight (LoS) links, we propose a solution to support the stranded users using the RIS-aided HAPS. Particularly, we propose to support 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 supported 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 QoS and network resources, such as transmit power and the number of RIS reflecting units.