In this paper, we propose a minimum adjustment consensus framework for the social network group decision-making (SN-GDM) with incomplete linguistic preference relations (ILPRs). The extant studies ignore the influence of network structure on the decision-makers’ (DMs’) weights, and set a fixed parameter to adjust DM’s preferences that may lead to the inefficiency of reaching a consensus. To solve these issues, we first propose a weight allocation method with the structural hole theory by analyzing the tie strength and topology structure of DM’s social networks. After obtaining DMs’ weights, the consistency/consensus indexes at three levels are constructed and used to identify the inconsistent DMs. Then, a novel minimum adjustment consensus model (MACM) for ILPRs is proposed to obtain the optimal adjustment parameters, which are used to recommend customized adjustments in the feedback mechanism. The existence of optimal solutions and the convergence of the proposed consensus models under certain conditions are also proved. Finally, the validity of the proposed method is verified by an application example. Different from the extant MACMs, we optimized the adjustment parameters just for inconsistent DMs instead of all DMs’ adjusted preference values. With less number of consensus rounds and lower costs, we also improved the classical feedback mechanism and established its connection with the current MACMs.