It is now well-known that automatic speaker verification (ASV) systems can be spoofed using various types of adversaries. The usual approach to counteract ASV systems against such attacks is to develop a separate spoofing countermeasure (CM) module to classify speech input either as a bonafide, or a spoofed utterance. Nevertheless, such a design requires additional computation and utilization efforts at the authentication stage. An alternative strategy involves a single monolithic ASV system designed to handle both zero-effort imposter (non-targets) and spoofing attacks. Such spoof-aware ASV systems have the potential to provide stronger protections and more economic computations. To this end, we propose to generalize the standalone ASV (G-SASV) against spoofing attacks, where we leverage limited training data from CM to enhance a simple backend in the embedding space, without the involvement of a separate CM module during the test (authentication) phase. We propose a novel yet simple backend classifier based on deep neural networks and conduct the study via domain adaptation and multi-task integration of spoof embeddings at the training stage. Experiments are conducted on the ASVspoof 2019 logical access dataset, where we improve the performance of statistical ASV backends on the joint (bonafide and spoofed) and spoofed conditions by a maximum of 36.2% and 49.8% in terms of equal error rates, respectively.