Auction-based Federated Learning (AFL) has emerged as an important research field in recent years. The prevailing strategies for FL data consumers (DCs) assume that the entire team of the required data owners (DOs) for an FL task must be assembled before training can commence. In practice, a DC can trigger the FL training process multiple times. DOs can thus be gradually recruited over multiple FL model training sessions. Existing bidding strategies for AFL DCs are not designed to handle such scenarios. Therefore, the problem of multi-session AFL remains open. To address this problem, we propose the Multi-session Budget Optimization Strategy for forward Auction-based Federated Learning (MBOS-AFL). Based on hierarchical reinforcement learning, MBOS-AFL jointly optimizes inter-session budget pacing and intra-session bidding for AFL DCs, with the objective of maximizing the total utility. Extensive experiments on six benchmark datasets show that it significantly outperforms seven state-of-the-art approaches. On average, \methodname{} achieves 12.28% higher utility, 14.52% more data acquired through auctions for a given budget, and 1.23% higher test accuracy achieved by the resulting FL model compared to the best baseline. To the best of our knowledge, it is the first budget optimization decision support method with budget pacing capability designed for DCs in multi-session forward auction-based FL.