Lee et al. proposed a new bootstrapping algorithm based on homomorphic automorphism, which merges the empty sets of ciphertexts by adjusting the window size. This algorithm supports arbitrary secret key distributions with no additional runtime costs while using small evaluation keys. However, our implementation reveals that once the window size exceeds a certain threshold, the time required for bootstrapping remains relatively constant. This observation prompts the question of how to further reduce the running time. To address this challenge, we introduce a new trick called Adaptive Key Update (AKU). With AKU and automorphism techniques, we propose a new bootstrapping algorithm for Gaussian secret keys that requires only n external products and no switching for blind rotation. Building on this, we employ window size optimization and key switching techniques to further improve the algorithm. The improved algorithm provides a useful trade-off between key storage and computational efficiency, depending on the choice of the window size. Compared to the current fastest FHEW bootstrapping method for Gaussian secret keys (the LLWW+ method proposed by Li et al.), our AKU-based algorithm reduces the number of key switching by 76% and decreases the running time of bootstrapping by 20.7%. At this point, the practical runtime for bootstrapping is approximately equal to that of performing only n external products.