The problem of packing equal spheres in a spherical container is a classic global optimization problem, which has attracted enormous studies in academia and found various applications in industry. This problem is computationally challenging, and many efforts focus on small-scale instances with the number of spherical items less than 200 in the literature. In this work, we propose an efficient local search heuristic algorithm named solution space exploring and descent for solving this problem, which can quantify the solution's quality to determine the number of exploring actions and quickly discover a high-quality solution. Besides, we propose an adaptive neighbor object maintenance method to speed up the convergence of the continuous optimization process and reduce the time consumption. Computational experiments on a large number of benchmark instances with $5 \leq n \leq 400$ spherical items show that our algorithm significantly outperforms the state-of-the-art algorithm. In particular, it improves the 274 best-known results and matches the 84 best-known results out of the 396 well-known benchmark instances.

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