The battery energy storage system (BESS) has immense potential for enhancing grid reliability and security through its participation in the electricity market. BESS often seeks various revenue streams by taking part in multiple markets to unlock its full potential, but effective algorithms for joint-market participation under price uncertainties are insufficiently explored in the existing research. To bridge this gap, we develop a novel BESS joint bidding strategy that utilizes deep reinforcement learning (DRL) to bid in the spot and contingency frequency control ancillary services (FCAS) markets. Our approach leverages a transformer-based temporal feature extractor to effectively respond to price fluctuations in seven markets simultaneously and helps DRL learn the best BESS bidding strategy in joint-market participation. Additionally, unlike conventional "black-box" DRL model, our approach is more interpretable and provides valuable insights into the temporal bidding behavior of BESS in the dynamic electricity market. We validate our method using realistic market prices from the Australian National Electricity Market. The results show that our strategy outperforms benchmarks, including both optimization-based and other DRL-based strategies, by substantial margins. Our findings further suggest that effective temporal-aware bidding can significantly increase profits in the spot and contingency FCAS markets compared to individual market participation.