The NIST standardization process for post-quantum cryptography has been drawing the attention of researchers to the submitted candidates. One direction of research consists in implementing those candidates on embedded systems and that exposes them to physical attacks in return. The Classic McEliece cryptosystem, which is among the four finalists of round 3 in the Key Encapsulation Mechanism category, builds its security on the hardness of the syndrome decoding problem, which is a classic hard problem in code-based cryptography. This cryptosystem was recently targeted by a laser fault injection attack leading to message recovery. Regrettably, the attack setting is very restrictive and it does not tolerate any error in the faulty syndrome. Moreover, it depends on the very strong attacker model of laser fault injection, and does not apply to optimised implementations of the algorithm that make optimal usage of the machine words capacity. In this article, we propose a to change the angle and perform a message-recovery attack that relies on side-channel information only. We improve on the previously published work in several key aspects. First, we show that sidechannel information, obtained with power consumption analysis, is sufficient to obtain an integer syndrome, as required by the attack framework. This is done by leveraging classic machine learning techniques that recover the Hamming weight information very accurately. Second, we put forward a computationallyefficient method, based on a simple dot product and informationset decoding algorithms, to recover the message from the, possibly inaccurate, recovered integer syndrome. Finally, we present a masking countermeasure against the proposed attack.