The existing offset-aware response-time analysis of Controller Area Network (CAN) for mixed messages has certain practical limitations. It is based on the assumption that the jitter and deadline of a message are smaller or equal to the transmission period. However, practical systems may contain messages with release jitter greater than the period. Consequently, the deadlines specified for such messages are also greater than their periods. In this paper, we extend the existing response-time analysis for mixed messages in CAN that are scheduled with offsets and have arbitrary jitter and deadlines. Mixed messages are implemented by several higher-level protocols based on CAN that are used in the automotive industry. The extended analysis is applicable to any higher-level protocol for CAN that uses periodic, sporadic and mixed transmission modes.