In the present work, we extend the semi-cyclic contraction conditions established by Gabeleh and Abkar [9] by introducing the concept of a $p$-semi-cyclic contraction for a pair of mappings $(S, T)$ on the union of $p(\geq 2)$ nonempty subsets in a Banach space. In the sequel, we establish several fundamental results, including existence and convergence theorems for best proximity points corresponding to the pair $(S,T)$. Additionally, illustrative examples are provided to support our findings.