We investigate the problem of detecting the distributed targets buried in the Gaussian noise whose covariance matrix is unknown when signal mismatch occurs. The idea is to add a fictitious signal under the null hypothesis of the origin detection problem so that when signal mismatch occurs, the fictitious signal captures the mismatched signals, thus making the null hypothesis more plausible. More precisely, the fictitious signal is modeled as a Gaussian component with a covariance matrix of a stochastic factor multiplied by a rank-one matrix. The generalized likelihood ratio test (GLRT) is employed to address the modification detection problem. We present an exhaustive derivation of the detector and prove that it possesses the constant false alarm rate (CFAR) property. The performance analysis demonstrates the effectiveness of the proposed detector. When the SNR is 23 dB, as generalized cosine squared decreases from 1 to 0.83, the detection probability of the proposed GLRT-SL drops to 0.65, exhibiting the fastest decline compared to the G-ABORT-HE, which falls to 0.98, and the GW-ABORT-HE, which decreases to 0.85.