The existing tamper detection schemes for absolute moment block truncation coding (AMBTC) compressed images are able to detect the tampering. However, the marked image qualities of these schemes can be enhanced, and their authentication methods may fail to detect some special tampering. We propose a secure AMBTC tamper detection scheme that preserves high image fidelity with excellent detectability. In the proposed approach, a bit in bitmaps of AMBTC codes is sequentially toggled to generate a set of authentication codes. The one that causes the least distortion is embedded into the quantization levels with the guidance of a key-generated reference table (RT). Without the correct key, the same reference table cannot be constructed. Therefore, the proposed method is able to detect various kinds of malicious tampering, including those special tampering techniques designed for RT-based authentication schemes. The proposed method not only offers better image quality, but also provides an excellent and satisfactory detectability as compared with previous works.