This letter focuses on the design of selective receivers for homogeneous scenarios where a very small number of secondary data are available. To this end, at the design stage it is assumed that the cell under test (CUT) contains a fictitious signal orthogonal to the nominal steering vector under the null hypothesis; the clutter covariance matrix is modeled as a random matrix with an inverse complex Wishart distribution. Under the above assumptions, we devise two Bayesian detectors based on the GLRT criterion, both one-step and two-step. It is shown that the proposed detectors have the same detection structure as their non-Bayesian counterparts, substituting the colored diagonal sample covariance matrix (SCM) for the classic one. Finally, a performance assessment, conducted by Monte Carlo simulations, has shown that our detectors ensure better rejection capabilities of mismatched signals than the existing Bayesian detectors, at the price of a certain loss in terms of detection of matched signals.