Impossible differential cryptanalysis and impossible polytopic cryptanalysis are among the most effective techniques for evaluating the security of block ciphers. However, previous automatic search methods for their distinguishers—dimpossible differentials and impossible polytopic transitions—neither account for the influence of the key schedule in single-key settings nor are applicable to block ciphers featuring large S-boxes, variable rotations, or key-dependent permutations. Furthermore, existing approaches fail to search for clusters of impossible differentials when all details of a block cipher are considered. In contrast to previous methods that focus solely on the propagation of differences or s-difference, we redefine impossible differentials and impossible (s+ 1)-polytopic transitions based on state propagation. This redefinition enables us to overcome the limitations inherent in earlier methodologies. Theoretically, we demonstrate that traditional definitions of impossible differentials and impossible (s+ 1)-polytopic transitions correspond to subsets of our redefined concepts, which offer broader analytical perspectives. Technically, we reformulate the automatic search model and develop an SAT-based tool to efficiently evaluate our redefined impossible differentials and impossible (s+ 1)-polytopic transitions. Building upon this foundational search method, we construct a comprehensive framework for detecting clusters of impossible differentials and impossible (s+1)-polytopic transitions. This framework not only fully incorporates the details and differential properties of block ciphers but is also applicable to those employing large S-boxes while considering the full linear layer. As a result, we derive new impossible differentials for GIFT64, PRINTcipher48/96, MISTY1, RC5-32/64/128 and SPECK, as well as new clusters of impossible differentials for SPECK, DES and ARIA. In assessing resistance against impossible differentials, we apply our method to evaluate the security of GIFT64, PRINTcipher48/96, MISTY1, SPECK, SIMON, and DES while accounting for all details of the block ciphers. Moreover, we propose acceleration strategies and apply them to evaluate the security of MISTY1 and AES-128. Notably, we prove that no 5-round impossible differentials with one active input byte and one active output byte exist for AES-128, even when considering the dependencies among three consecutive round keys. Finally, in exploring new impossible (s+1)-polytopic transition, we apply our approach to PRINTcipher48, GIFT64, RC5-32/64 and SIMON32-64, successfully yielding the corresponding distinguishers for the first time.