Accurate and reliable phase-resolved prediction of ocean surface waves is crucial for many offshore operations in ocean engineering and marine science. One important application is in optimal control of a power take-off in a wave energy converter, leading to significantly higher power production. Our interest is forecasting wave fields based on measurements obtained from multiple upwave locations in moderate seas with small directional spreading angles, as occurs along the south coast of Australia. The prediction model, based on FFTs and propagation of waves according to the linear dispersion relation, is applied to both wave groups and irregular wave fields generated in a wave basin and, additionally, to ocean waves measured with drifting wave buoys. To account for spreading, the model numerically advances linear, plane (i.e., long-crested) waves in space at an optimum offset angle equal to the underlying sea-state root mean square spreading angle. Averaging predictions from a few slightly separated measurement locations, each weighted according to its estimated variance, results in more accurate predictions than from any single location. We also assess in detail the effect of drifting buoy measurements in both long-crested and short-crested seas using synthetic wave records and show that it is possible to satisfactorily reconstruct the signal at fixed points based on the Doppler shift felt by the drifting buoy. The reconstructed signals give much better predictions compared to those completely neglecting the effect of even rather slow drift due to current.