Motivated by recent developments of Kil et al., who proposed a new class of composed $S$-metric type spaces extending the notion of controlled $S$-metric spaces, this paper investigated fixed point phenomena within this broader setting. We derived existence and uniqueness theorems for fixed points corresponding to various contractive conditions by making essential use of the structural features of the proposed framework. The presented theorems encompassed and extended a number of well-established results in the existing literature. To demonstrate the practical relevance of the theory, a concrete example was included, illustrating the effectiveness of composed $S$-metric spaces in addressing the solvability of $n$th-degree polynomial equations.