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Promoting cooperation in social dilemma games, particularly in multi-player settings, is a challenge with important implications for real-world systems. While many mechanisms exist to foster cooperation in iterated two-player games, the dynamics of cooperation in $n$-player social dilemmas are comparatively less understood. Techniques such as tagging, group selection, and pool rewarding have been applied to $n$-player games, but these approaches often rely on unrealistic assumptions and result in suboptimal cooperation levels. We present an evolutionary approach to engendering cooperation in the $n$-player Snowdrift game. Our hybrid method combines tagging with tournament selection to evolve individual strategies while utilizing group selection mechanisms for dynamic group restructuring. We evaluate the efficacy of this combined approach across varying cost-benefit ratios, population sizes, and group restructuring schemes. Experimental results show that our model consistently promotes and sustains high levels of cooperation in the $n$-player Snowdrift game. This work provides valuable insights into scalable cooperation mechanisms in multi-agent systems facing social dilemmas.