Off-site steel construction requires close coordination among fabrication, transportation, intermediate yard operations, site delivery, and erection. Because steel components are heavy, project-specific, and sequence-dependent, early delivery may cause site congestion and re-handling, whereas late delivery may lead to crane idle time and erection delay. This study proposes a hybrid Stackelberg-cooperative game framework for fourth-party logistics (4PL) coordination in off-site steel construction supply chains. A mixed-integer model is developed to coordinate the main operational decisions of the off-site steel construction supply chain. The model links fabrication assignment and carrier selection with delivery routing, yard utilization, site storage, and erection planning, while ensuring that capacity limits, delivery windows, kitting requirements, crane availability, laydown restrictions, and precedence relations are satisfied. The Stackelberg layer designs incentive-compatible service fees, bonuses, penalties, and risk-sharing parameters, while the cooperative layer allocates the coordination surplus using the Shapley value. An Iranian project-scale case study shows that the proposed mechanism reduces total supply chain cost by 14.2%, increases the on-time delivery ratio from 54.8% to 88.9%, and decreases crane idle time from 79 to 19 hours compared with the decentralized baseline. The results also indicate substantial reductions in delayed components, site congestion, and re-handling compared with traditional 3PL and non-incentive 4PL coordination.