Automatic modulation classification (AMC) is an important function in wireless communication systems that is used to identify the modulation type of a signal without prior knowledge. AMC has historically been done using likelihood or feature-based methods, yet recent research has focused on using deep neural networks (DNNs) as they outperform the classical methods in challenging signal channel conditions. However, deep learning (DL) based classifiers are vulnerable to adversarial attacks that can significantly deteriorate their classification performance. This paper explores the robustness of different AMC classifiers to the white-box fast gradient method (FGM) and projected gradient descent (PGD) attacks under different perturbation-to-noise ratios (PNRs) and signal-to-noise ratios (SNRs) for a noisy signal channel. The investigated AMC classifiers consist of the quasi-hybrid likelihood ratio test (QHLRT), a k-nearest neighbour (KNN) that uses higher-order cumulants, and the parameter estimation and transformation-based CNN-GRU deep neural network (PET-CGDNN). The adversarial attacks are found to have limited transferability to the QHLRT and the KNN classifiers when scaled to be imperceptible against the noise of the signal channel. Based on this finding, we propose a hybrid classifier that uses the neural rejection technique through a support vector machine (SVM) that acts as a switching mechanism to decide whether to use the KNN or PET-CGDNN to classify the modulation type. The hybrid classifier demonstrates improved robustness against attacks, while benefiting from the good performance of the DNN.