For self-organized networks that possess highly decentralized and self-organized natures, neither the identity nor the number of processes is known to all participants at the beginning of the computation because no central authority exists to initialize each participant with some context information. Hence, consensus, which is essential to solving the agreement problem, in such networks cannot be achieved in the ways for traditional fixed networks. To address this problem of Consensus with Unknown Participants (CUP), a variant of the traditional consensus problem was proposed in the literature, by relaxing the requirement for the original knowledge owned by every process about all participants in the computation. Correspondingly, the CUP problem considering process crashes was also introduced, called the Fault-Tolerant Consensus with Unknown Participants (FT-CUP) problem. In this paper, we propose a knowledge connectivity condition sufficient for solving the FT-CUP problem, which is improved from the one proposed in our previous work.