Based on new log-t-based detectors, we propose to improve the detection performances of the log-t-Constant False Alarm Rate (log-t-CFAR) detector for a non-homogeneous Weibull background. This paper is twofold. We first resort to the Automatic Constant False Censoring Rate (CFCR) algorithm, which guarantees an accurate rejection of an a priori unknown number of outliers. That is, we introduce two hybrid detectors by coupling the log-t-CFAR algorithm to the Maximum Likelihood-CFCR (MLE-CFCR) algorithm, yielding the H-MLE/log-t-CFAR detector, and to the Weber-Haykin Constant False Censoring Rate (WH-CFCR) algorithm, yielding the H-WH/log-t-CFAR detector. Then, based on the Variability Index (VI) as a background discriminator, we propose the Switching VI-log-t-CFAR (SVI-log-t-CFAR) detector. Thus, depending on the background heterogeneity, this detector has the capability to switch automatically to the appropriate detector; namely, the log-t-CFAR detector, in case of a homogeneous background, either one of the hybrid detectors, in case of the presence of outliers or the Automatic Edge Censoring log-t-CFAR (AEC-log-t-CFAR) detector, in case of the presence of a clutter edge. We assess the efficiency of these detectors through intensive Monte Carlo simulations. We show that, while no additional detection performances are observed in a homogeneous background, the new detectors exhibit a significant CFAR gain with respect to the log-t-CFAR detector in the presence of any inhomogeneity within the reference window.