Critical learning periods (CLPs) in federated learning (FL) refer to early stages during which low-quality contributions (e.g., sparse training data availability) can permanently impair the performance of the global model owned by the cloud server. However, existing incentive mechanisms typically assume temporal homogeneity, treating all training rounds as equally important, thereby failing to prioritize and attract high-quality contributions during CLPs. This inefficiency is compounded by information asymmetry due to privacy regulations, where the cloud lacks knowledge of client training capabilities, leading to adverse selection and moral hazard. Thus, in this article, we propose a time-aware contract-theoretic incentive framework, named Right Reward Right Time (R3T), to encourage client involvement, especially during CLPs, to maximize the utility of the cloud server. We formulate a cloud utility function that captures the trade-off between the achieved model performance and rewards allocated for clients' contributions, explicitly accounting for client heterogeneity in time and system capabilities, effort, and joining time. Then, we devise a CLP-aware incentive mechanism deriving an optimal contract design that satisfies individual rationality, incentive compatibility, and budget feasibility constraints, motivating rational clients to participate early and contribute efforts. By providing the right reward at the right time, our approach can attract the highest-quality contributions during CLPs. Simulation and proof-of-concept studies show that R3T mitigates information asymmetry, increases cloud utility, and yields superior economic efficiency compared to conventional incentive mechanisms. Our proof-of-concept results demonstrate up to a 47.6% reduction in the total number of clients and up to a 300% improvement in convergence time while achieving competitive test accuracy.