There is a growing literature studying return spillovers between similar assets and assets of different classes during crisis periods. However, less is known about return spillovers across stock sectors under high and low volatility regimes and whether they are affected by oil price volatility. Using daily data from May 10th, 2007 to February 28th, 2020, we first study the return spillovers between US stock sectors under low and high volatility regimes by implementing a Markov regime-switching vector autoregression with exogenous variables model, while considering the Fama-French factors as conditioning variables. Return spillovers under low and high volatility regimes show that the energy sector is the largest transmitter and receiver of spillovers to/from other US equity sectors. Rolling window analysis shows that spillovers intensified since the outbreak of the COVID19 pandemic. Second, we apply linear and non-linear Granger causality tests from oil price volatility to the spillover indices. The results show evidence that oil volatility has a causal impact on the spillover dynamics of US stock sectors and that the impact is particularly strong in the high volatility regime. Although the energy sector is one of the smallest sectors of the US stock market, it plays a large role in the network connectedness of stock sectors. The results are of interest to individual and institutional investors who consider US equity investments and to policymakers.