The future Nankai Trough tsunami is one of the imminent threats to the Japanese coastal communities that could potentially cause a catastrophic event. As a part of the countermeasure efforts for such an occurrence, this study analyzes the efficacy of combining tsunami data assimilation (DA) and waveform inversion (WI). The DA is used to continuously refine a wavefield model whereas the WI is used to estimate the tsunami source. We consider a future scenario of the Nankai Trough tsunami recorded at various observational systems, including ocean bottom pressure (OBP) gauges, global positioning system (GPS) buoys, and ship height positioning data. Since most of the OBP gauges are located inside the source region, the recorded tsunami signals exhibit significant offsets from surface measurements due to coseismic seafloor deformation effects. Such biased data are not applicable to the standard DA, but can be taken into account in the WI. On the other hand, the use of WI for the ship data may not be practical because a considerably large precomputed tsunami database is needed to cope with the spontaneous ship locations. The DA is more suitable for such an observational system as it can be executed sequentially in time and does not require precomputed scenarios. Therefore, the combined approach of DA and WI allows us to concurrently make use of all observational resources. Additionally, we introduce a bias correction scheme for the OBP data to improve the accuracy, and an adaptive thinning of observations to determine the efficient number of observations.