<p>We investigate the average spread complexity of a generic two-level subsystem embedded in a larger system to assess the influence of energy-level statistics, comparing chaotic and integrable systems. Focusing first on nearest-neighbor level spacings, we observe the characteristic slope-dip-ramp-plateau structure. Further analysis shows that certain matrix models exhibit additional peaks prior to saturation in some special scenarios, which motivates a generalization to higher-order level spacings. While this structure persists in chaotic systems, we find that integrable systems can also display similar features, underscoring limitations of spread complexity as a universal diagnostic of quantum chaos for arbitrary states.</p>