Enhancing future economic growth through higher economic complexity is vital for sustainable growth strategies. Even though most studies are concerned with the key factors governing the economic complexity process, there needs to be more studies about the impact of risk factors on economic sophistication evolvement, particularly for oil-rentier economies. The present paper intends to delve into the dynamic connectedness between some risk factors, such as economic, political, and geopolitical risks and oil price volatility and the progress of economic complexity. We utilized quarterly data (1995Q1-2021Q4) for Saudi Arabia as a heavy oil-rich economy. As the first study to delve into the short-long term connections at various time scales and frequencies within a multivariable setting, we contribute to the literature by offering a spotless picture of economic complexity risk factors. In doing so, we resort to a novel wavelet local multiple correlation method, which can explore the varying patterns of time periods in the interconnections between the variables. Our findings disclose that oil volatility, geopolitical risk, and global uncertainties are positively connected to economic complexity over the long run. In contrast, their short-term effect is weak and mostly insignificant. These results indicate the resiliency of the Saudi economy to external to oil volatility shocks and intensification of global and local uncertainties. These outcomes offer policymakers new insights and prominent policy recommendations when designing economic strategies to achieve higher economic sophistication and sustainable growth.