In the post-pandemic social and economic conditions, the proper assessment of the effectiveness of regional innovation systems (RISs) becomes a key endeavor. In our paper, we highlight the necessity to enhance the set of indicators used to evaluate the performance of regional innovation systems in countries with varying innovation capabilities. Specifically, we concentrate on examining case studies from the United States, Japan, China, and the Czech Republic, comparing their experiences with the current situation to innovations and innovation systems in Russia and drawing lessons for this country. Utilizing the Global Innovation Index (GII) rankings, we conduct an analysis of the characteristics of innovative progress and propose specific groups of indicators that can enhance the effectiveness of evaluating the innovative advancement of different regions. Moreover, we determine the need for uniqueness, flexibility, and adaptability of these based on the state’s strategic guidelines in the field of innovation and the innovative potential of the territory as well as the factors of external and internal influence. In addition, we conduct and present the results of the bibliometric network analysis of the research publications retrieved from the Web of Science (WoS) database using VOSViewer software and covering the role of regional innovation systems (RISs) in shaping up the national innovation systems (NISs) both in general terms and specifically applied to the case of Russia. Our results might be relevant for the stakeholders and policymakers who are engaged in promoting innovation, regional development, and sustainable economic growth, as well as for the academics working on the topics of innovation and economic development.