Localizing gas sources is a challenging task due to the complex nature of gas dispersion. Informative Path Planning (IPP) plays a crucial role in guiding robots to sample at high-information positions, thereby accelerating the estimation process. Existing probabilistic gas source localization methods often require robots to halt at sampling positions, averaging gas measurements over time. Consequently, when selecting the next sampling position, information gains are usually computed precisely through computationally heavy procedures, limiting evaluations to a small set of potential positions. In our previous work, we introduced a sense-inmotion strategy that eliminates the need for prolonged stops at sampling points, therefore allowing the incorporation of measurements taken during robot movement. Building upon this advancement, we propose to extend information gain evaluation in a more continuous manner, from a point evaluation to a path evaluation. However, existing IPP methods are too computationally expensive when transitioning from goal-based to region-based evaluations. To address this challenge, we first assess three lightweight information extraction metrics. Based on the selected metrics, we propose a novel IPP algorithm that computes cumulative information gain along the robot's path and dynamically prioritizes exploration or exploitation based on the uncertainty of the source estimation. The proposed method is extensively evaluated through both high-fidelity simulations and physical experiments. Results show that our proposed method consistently outperforms a benchmark state-of-theart method, achieving a 40% increase in source localization success rate and halving the experimental time in challenging environments.