Side-channel attacks are a major threat to the security of cryptographic implementations, particularly for small devices that are under the physical control of the adversary. While several strategies for protecting against side-channel attacks exist, these often fail in practice due to unintended interactions between values deep within the CPU. To detect and protect from side-channel attacks, several automated tools have recently been proposed; one of their common limitations is that they only support first-order leakage. In this work, we present the first automated tool for detecting and eliminating higher-order leakage from cryptographic implementations. Rosita++ proposes statistical and software-based tools to allow high-performance higher-order leakage detection. It then uses the code rewrite engine of Rosita (Shelton et al. NDSS 2021) to eliminate detected leakage. For the sake of practicality we evaluate Rosita++ against second and third order leakage, but our framework is not restricted to only these orders. We evaluate Rosita++ against second-order leakage with three-share implementations of two ciphers, PRESENT and Xoodoo, and with the second-order Boolean-to-arithmetic masking, a core building block of masked implementations of many cryptographic primitives, including SHA-2, ChaCha and Blake. We show effective second-order leakage elimination at a performance cost of 36% for Xoodoo, 189% for PRESENT, and 29% for the Boolean-to-arithmetic masking. For third-order analysis, we evaluate Rosita++ against the third-order leakage using a four-share synthetic example that corresponds to typical four-share processing. Rosita++ correctly identified this leakage and applied code fixes.