Interest in delivering cellular communication using a high-altitude platform (HAP) is increasing partly due to its wide coverage capability. In this paper, we formulate analytical expressions for estimating the area of a HAP beam footprint, average per-user capacity per cell, average spectral efficiency (SE) and average area spectral efficiency (ASE), which are relevant for radio network planning, especially within the context of HAP extended contiguous cellular coverage and capacity. To understand the practical implications, we propose an enhanced and validated recursive HAP antenna beam-pointing algorithm, which forms HAP cells over an extended service area while considering beam broadening and the degree of overlap between neighbouring beams. The performance of the extended contiguous cellular structure resulting from the algorithm is compared with other alternative schemes using the carrier-to-noise ratio (CNR) and carrier-to-interference-plus-noise ratio (CINR). Results show that there is a steep reduction in average ASE at the edge of coverage. The achievable coverage is limited by the minimum acceptable average ASE at the edge, among other factors. In addition, the results highlight that efficient beam management can be achieved using the enhanced and validated algorithm, which significantly improves user CNR, CINR, and coverage area compared with other benchmark schemes. A simulated annealing comparison verifies that such an algorithm is close to optimal.