Side-channel attacks exploit information leaked through non-primary channels, such as power consumption, electromagnetic emissions, or timing, to extract sensitive data from cryptographic devices. Over the past three decades, side-channel analysis has evolved into a mature research field with well-established methodologies for analyzing standard cryptographic algorithms like the Advanced Encryption Standard (AES). However, the integration of side-channel analysis with formal methods remains relatively unexplored. In this paper, we present a hybrid attack on AES that combines side-channel analysis with SAT. We model AES as a SAT problem and leverage hints of the input and output values of the S-boxes, extracted via profiled deep learning-based power analysis, to solve it. Experimental results on an ATXmega128D4 MCU implementation of AES-128 demonstrate that the SAT-assisted approach consistently recovers the full encryption key from a single trace, captured from devices different from those used for profiling, within one hour. In contrast, without SAT assistance, the success rate remains below 80% after 26 hours of key enumeration.