Blockchain has brought great potentials in improving Space-Air-Ground Integrated Networks (SAGINs) in terms of security and efficiency. In blockchain-integrated SAGINs, many applications and services inherently require both the communication contents and communication behaviors to be secure against eavesdroppers, in which a covert communication algorithm is always deployed as a fundamental communication component. However, existing covert communication schemes suffer from critical problems. On the one hand, they require a sender to locally maintain a cryptographic key for a long period of time, which is very costly and inefficient to renew the security protection. On the other hand, the ciphertext of covertly-sent data would explicitly appear in the network, and thereby the schemes are vulnerable to secret key breach. In this paper, we propose a secure and efficient covert communication scheme for blockchain-integrated SAGINs, dubbed CC-BSAGINs, to free the sender from maintaining secret keys. The key technique is to map the covertly-sent data to some transactions on the underlying blockchain in a secure and efficient way; the mapping information is sent via a covert communication algorithm. Such a two-step mechanism releases the sender from key management and does not require the ciphertext to be communicated. We provide formal security proofs and conduct a comprehensive performance evaluation, which demonstrates the security and efficiency of CC-BSAGINs.