With recent advances in location-based services, satellite navigation systems that guarantee high precision and high integrity have become increasingly important. The Japanese Quasi-Zenith Satellite System (QZSS) provides the Centimeter-Level Augmentation Service (CLAS) via the L6 signal. However, unlike standard QZSS navigation messages such as LNAV and CNAV, CLAS augmentation messages currently lack an authentication mechanism and are consequently vulnerable to spoofing attacks. Furthermore, since CLAS is already an operational service, incorporating authentication by modifying the existing message specification is not possible. To address this issue, in 2024 Jeon et al. proposed an authentication method that uses the reserved field in the navigation messages of the L62 signal. However, this method requires significant Time To Authentication (TTA) to deal with multiple fragments of an Elliptic Curve Digital Signature Algorithm (ECDSA) signature transmitted over multiple frames. In this paper, we propose a novel authentication protocol to mitigate this issue. The proposed protocol exploits the property of the ECDSA signature pair (r, s), where the r component can be generated independently of the message content. By transmitting r concurrently with the data, the receiver can initiate signature verification earlier. Analytical results demonstrate that the proposed protocol reduces TTA by up to 25% compared to the previous method. Experimental results on both a PC and a Raspberry Pi confirm that the computational overhead of the proposed scheme is negligible.