Image steganography is the process of hiding secret data in a cover image by subtle perturbation. Recent studies show that it is feasible to use a fixed neural network for data embedding and extraction. Such Fixed Neural Network Steganography (FNNS) demonstrates favorable performance without the need for training networks, making it more practical for real-world applications. However, the stego-images generated by the existing FNNS methods exhibit high distortion, which is prone to be detected by steganalysis tools. To deal with this issue, we propose a Cover-separable Fixed Neural Network Steganography, namely Cs-FNNS. In Cs-FNNS, we propose a Steganographic Perturbation Search (SPS) algorithm to directly encode the secret data into an imperceptible perturbation, which is combined with an AI-generated cover image for transmission. Through accessing the same deep generative models, the receiver could reproduce the cover image using a pre-agreed key, to separate the perturbation in the stego-image for data decoding. such an encoding/decoding strategy focuses on the secret data and eliminates the disturbance of the cover images, hence achieving a better performance. We apply our Cs-FNNS to the steganographic field that hiding secret images within cover images. Through comprehensive experiments, we demonstrate the superior performance of the proposed method in terms of visual quality and undetectability. Moreover, we show the flexibility of our Cs-FNNS in terms of hiding multiple secret images for different receivers.