Adaptive track initiation remains a crucial component of many modern multi-target tracking systems. For labeled random finite sets multi-object filters, prior work has been established to construct a labeled multi-object birth density using measurements from multiple sensors. A naive construction of this adaptive birth set density results in an exponential number of newborn components in the number of sensors. A truncation procedure was provided that leverages a Gibbs sampler to truncate the birth density, reducing the complexity to quadratic in the number of sensors. However, only a limited discussion has been provided on additional algorithmic techniques that can be employed to substantially reduce the complexity in practical tracking applications. In this paper, we propose five efficiency enhancements for the labeled random finite sets multi-sensor adaptive birth procedure. Simulation results are provided to demonstrate their computational benefits and show that they result in a negligible change to the multi-target tracking performance.