The construction industry is undergoing a paradigm shift driven by the growing demand for greater flexibility, stakeholder engagement, and sustainable outcomes. Traditional project management (TPM) models, though universally used, do not offer adequate flexibility in changing situations. Agile project management (APM), being iterative and customer-centric, is a suitable choice—especially when combined with Lean and Hybrid approaches. This review synthesizes an extensive set of peer-reviewed papers from 2010 to 2024 to evaluate the contextual appropriateness, advantages, and disadvantages of Agile, Lean, Traditional, and Hybrid approaches in the construction industry. Major conclusions show Agile as ideal for planning and design phases, Lean for implementation, and TPM for governance and regulatory compliance. Phase-based hybrid models following these models per project phase have the most potential. The research suggests a four-phase integration model for organizational readiness, phase-based deployment, digital empowerment, and continuous feedback. It also outlines major enablers, such as leadership, technology integration, and training stakeholders, and prescribes barriers like cultural resistance, contractual inflexibility, and the absence of digital infrastructure. This paper adds a structured method to phased Agile implementation in construction, closing the gap between theoretical models and actual practice. It ends with tangible recommendations and areas of research—emphasizing longitudinal verification, industry-specific implementation, and incorporating sustainability—to further enhance future construction project delivery.