Since the outbreak of the COVID-19 pandemic, the SARS-CoV-2 coronavirus has accumulated an important amount of genetic and genomic variability through mutation and recombination events. To test evolutionary trends that could inform us on the adaptive process of the virus to its human host, we summarize all this sequence variability by computing the Sequence Compositional Complexity (SCC) in more than 23,000 high-quality coronavirus genome sequences from across the globe, covering the period spanning from the start of the pandemic in December 2019 to March 2022. In early samples, we found no statistical support for any trend in SCC values over time, although the virus as a whole appears to evolve faster than Brownian Motion expectation. However, in samples taken after the first Variant of Concern (VoC) with higher transmissibility (Alpha) emerges, and controlling for phylogenetic and sampling effects, we were able to detect a statistically significant trend for decreased SCC values over time. SARS-CoV-2 evolution towards lower values of genome heterogeneity is further intensified by the emergence of successive, widespread VoCs. Concomitantly to the temporal reduction in SCC, its absolute evolutionary rate kept increasing toward the present, meaning that the SCC decrease itself accelerated over time. As compared to Alpha or Delta variants, the currently dominant VoC, Omicron, shows much stronger trends in both SCC values and rates over time. These results indicate that the increases in fitness of variant genomes associated to a higher transmissibility leads to a reduction of their genome sequence heterogeneity, thus explaining the general slowdown of SCC along with the pandemic course.