Foundation models have been widely adopted as starting points for fine-tuning, leading to expert-level performance on various downstream tasks. They have been actively developed by improving the pretraining data and/or model architectures. As a result, whenever a new foundation model is released, we have to fine-tune it again on the downstream tasks, requiring non-negligible computational costs as well as data-management costs. A promising approach to this problem is inference-time model update, which modifies the prediction of the new pretrained model using those of the old pretrained/fine-tuned models. However, the existing methods have no theoretical guarantee of improvement in accuracy and may cause unexpected failure. In this paper, we establish the first method for inference-time model update with a theoretical guarantee of improvement in expected rewards, and show that our method can improve performance even when the existing methods fail theoretically. Finally, we empirically validate the effectiveness of our method in practical experiments.