In this paper, we focus on 1-bit precoding approaches for downlink massive multiple-input multiple-output (MIMO) systems, where we exploit the concept of constructive interference (CI). For both PSK and QAM signaling, we firstly formulate the optimization problem that maximizes the CI effect subject to the requirement of the 1-bit transmit signals. We then mathematically prove that, when employing the CI formulation and relaxing the 1-bit constraint, the majority of the transmit signals already satisfy the 1-bit formulation. Building upon this important observation, we propose a 1-bit precoding approach that further improves the performance of the conventional 1-bit CI precoding via a partial branch-and-bound (P-BB) process, where the BB procedure is performed only for the entries that do not comply with the 1-bit requirement. This operation allows a significant complexity reduction compared to the fully-BB (F-BB) process, and enables the BB framework to be applicable to the complex massive MIMO scenarios. We further develop an alternative 1-bit scheme through an `Ordered Partial Sequential Update' (OPSU) process that allows an additional complexity reduction. Numerical results show that both proposed 1-bit precoding methods exhibit a significant signal-to-noise ratio (SNR) gain for the error rate performance, especially for higher-order modulations.