As a critical threat to deep neural networks (DNNs), backdoor attacks can be categorized into two types, i.e., source-agnostic backdoor attacks (SABAs) and source-specific backdoor attacks (SSBAs). Compared to traditional SABAs, SSBAs are more advanced in that they have superior stealthier in bypassing mainstream countermeasures that are effective against SABAs. Nonetheless, existing SSBAs suffer from two major limitations. First, they can hardly achieve a good trade-off between ASR (attack success rate) and FPR (false positive rate). Besides, they can be effectively detected by the state-of-the-art (SOTA) countermeasures (e.g., SCAn). To address the limitations above, we propose a new class of viable source-specific backdoor attacks, coined as CASSOCK. Our key insight is that trigger designs when creating poisoned data and cover data in SSBAs play a crucial role in demonstrating a viable source-specific attack, which has not been considered by existing SSBAs. With this insight, we focus on trigger transparency and content when crafting triggers for poisoned dataset where a sample has an attacker-targeted label and cover dataset where a sample has a ground-truth label. Specifically, we implement $CASSOCK_{Trans}$ and $CASSOCK_{Cont}$. While both they are orthogonal, they are complementary to each other, generating a more powerful attack, called $CASSOCK_{Comp}$, with further improved attack performance and stealthiness. We perform a comprehensive evaluation of the three $CASSOCK$-based attacks on four popular datasets and three SOTA defenses. Compared with a representative SSBA as a baseline ($SSBA_{Base}$), $CASSOCK$-based attacks have significantly advanced the attack performance, i.e., higher ASR and lower FPR with comparable CDA (clean data accuracy). Besides, $CASSOCK$-based attacks have effectively bypassed the SOTA defenses, and $SSBA_{Base}$ cannot.