We quantify evolution in the cluster-scale stellar mass-halo mass (SMHM) relation's parameters using 2323 clusters and brightest central galaxies (BCGs) over the redshift range 0.03 ≤ z ≤ 0.60. The precision on the inferred SMHM parameters is improved by including the magnitude gap (m gap) between the BCG and fourth-brightest cluster member (M14) as a third parameter in the SMHM relation. At fixed halo mass, accounting for m gap, through a stretch parameter, reduces the SMHM relation's intrinsic scatter. To explore this redshift range, we use clusters, BCGs, and cluster members identified using the Sloan Digital Sky Survey C4 and redMaPPer cluster catalogs and the Dark Energy Survey redMaPPer catalog. Through this joint analysis, we detect no systematic differences in BCG stellar mass, m gap, and cluster mass (inferred from richness) between the data sets. We utilize the Pareto function to quantify each parameter's evolution. We confirm prior findings of negative evolution in the SMHM relation's slope (3.5σ), and detect negative evolution in the stretch parameter (4.0σ) and positive evolution in the offset parameter (5.8σ). This observed evolution, combined with the absence of BCG growth, when stellar mass is measured within 50 kpc, suggests that this evolution results from changes in the cluster's m gap. For this to occur, late-term growth must be in the intracluster light surrounding the BCG. We also compare the observed results to IllustrisTNG 300-1 cosmological hydrodynamic simulations and find modest qualitative agreement. However, the simulations lack the evolutionary features detected in the real data.