Previous research on no-reference (NR) quality assessment of multiply-distorted images focused mainly on three distortion types (white noise, Gaussian blur, and JPEG compression), while in practice images can be contaminated by many other common distortions due to the various stages of processing. Although MUSIQUE (MUltiply-and Singly-distorted Image QUality Estimator) Zhang et al., TIP 2018 is a successful NR algorithm, this approach is still limited to the three distortion types. In this paper, we extend MUSIQUE to MUSIQUE-II to blindly assess the quality of images corrupted by five distortion types (white noise, Gaussian blur, JPEG compression, JPEG2000 compression, and contrast change) and their combinations. The proposed MUSIQUE-II algorithm builds upon the classification and parameter-estimation framework of its predecessor by using more advanced models and a more comprehensive set of distortion-sensitive features. Specifically, MUSIQUE-II relies on a three-layer classification model to identify 19 distortion types. To predict the five distortion parameter values, MUSIQUE-II extracts an additional 14 contrast features and employs a multi-layer probability-weighting rule. Finally, MUSIQUE-II employs a new most-apparent-distortion strategy to adaptively combine five quality scores based on outputs of three classification models. Experimental results tested on three multiply-distorted and six singly-distorted image quality databases show that MUSIQUE-II yields not only a substantial improvement in quality predictive performance as compared with its predecessor, but also highly competitive performance relative to other state-of-the-art FR/NR IQA algorithms.