As privacy features in Android operating system improve, privacy-invasive apps may gradually shift their focus to non-standard and covert channels for leaking private user/device information. Such leaks also remain largely undetected by state-of-the-art privacy analysis tools, which are very effective in uncovering privacy exposures via regular HTTP and HTTPS channels. In this study, we design and implement, ThirdEye, to significantly extend the visibility of current privacy analysis tools, in terms of the exposures that happen across various non-standard and covert channels, i.e., via any protocol over TCP/UDP (beyond HTTP/S), and using multi-layer custom encryption over HTTP/S and non-HTTP protocols. Besides network exposures, we also consider covert channels via storage media that also leverage custom encryption layers. Using ThirdEye, we analyzed 12,598 top-apps in various categories from Androidrank, and found that 2887/12,598 (22.92%) apps used custom encryption/decryption for network transmission and storing content in shared device storage, and 2465/2887 (85.38%) of those apps sent device information (e.g., advertising ID, list of installed apps) over the network that can fingerprint users. Besides, 299 apps transmitted insecure encrypted content over HTTP/non-HTTP protocols; 22 apps that used authentication tokens over HTTPS, happen to expose them over insecure (albeit custom encrypted) HTTP/non-HTTP channels. We found non-standard and covert channels with multiple levels of obfuscation (e.g., encrypted data over HTTPS, encryption at nested levels), and the use of vulnerable keys and cryptographic algorithms. Our findings can provide valuable insights into the evolving field of non-standard and covert channels, and help spur new countermeasures against such privacy leakage and security issues.