The outbreak of the COVID-19 epidemic intensified the volatility of commodity markets (the energy and precious metals markets), which created a significant negative impact on the volatility spillovers among these markets. It may also have triggered a new volatility risk contagion. In this paper, we introduce the DCC-GARCH-CONNECTEDNESS approach to explore the volatility spillover level and multi-level spillover structure characteristics among the commodity markets before and during the COVID-19 epidemic in order to clarify the new volatility risk contagion patterns across the markets. The results implied several conclusions. (i) The COVID-19 epidemic has significantly improved the total volatility spillover level of the energy and precious metals markets and has enhanced the risk connectivity among the markets. (ii) The COVID-19 epidemic has amplified the volatility of the crude oil market, making it the main volatility spillover market, namely the source of volatility risk contagion. (iii) The COVID-19 epidemic outbreak enhanced the external risk absorption capacity of the natural gas and silver markets, and the absorption level of the external volatility spillover improved significantly. Furthermore, the risk absorption capacity of the gold market weakened, while the gold market has remained the endpoint of external volatility risk during the epidemic and has acted as a risk stabilizer. (iv) The volatility spillover among markets has clear time-varying characteristics and a positive connectedness with the severity of the COVID-19 epidemic. As the severity of the COVID-19 epidemic increases, the volatility risk connectivity among the markets rapidly increases.