Vehicular Ad-Hoc Networks (VANETs) have enhanced driver safety and traffic management efficiency, but the persistent challenges of privacy and communication overhead have garnered considerable attention. Conditional Privacy-Preserving Authentication (CPPA) ensures the anonymity of legitimate users in VANETs while also being able to trace the real identities of malicious users when necessary. However, existing CPPA schemes face challenges with key management, revocation, and generating significant overhead during frequent entity interactions. Blockchain-based CPPA (BCPPA) schemes offer a potential solution to these issues. Nevertheless, BCPPA sacrifices compatibility with practical applications, particularly in scenarios involving high-frequency entity interactions. It still requires separate transmission of sensitive and public information, posing communication overhead challenges in practical applications. To address this issue, this paper presents a pioneering cryptographic primitive for conditional anonymous authentication with secure communication in VANETs. We design a BCPPA with Out-of-band Communication scheme (BCPA-OC), which offers a unique capability to simultaneously support conditional anonymous authentication, and secure data transmission without any additional ciphertext overhead, to tackle the challenges posed during the communication process. Afterward, we introduce an application of BCPA-OC in cross-domain authentication. To showcase the superiority of our proposed BCPA-OC scheme, we refine the security model and rigorously prove its fulfillment of critical security properties, including integrity, non-repudiation, and indistinguishability.