Recently, in Diaz and Brevdo (J Fluid Mech 681: 567-596, 2011), further in the text referred to as D&B, we found an absolute/convective instability dichotomy at the onset of convection in a flow in a saturated porous layer with either horizontal or vertical solutal and inclined thermal gradients, and horizontal throughflow. The control parameter in D&B triggering the destabilization is the vertical thermal Rayleigh number, R-v. In this article, we treat the parameter cases considered in D&B in which the onset of convection has the character of convective instability and occurs through longitudinal modes. By increasing the vertical thermal Rayleigh number starting from its critical value, R-vc, we determine the value R-vt of R-v at which the transition from convective to absolute instability takes place and compute the physical characteristics of the emerging absolutely unstable wave packet. In some cases, the value of the transitional vertical thermal Rayleigh number, R-vt, is only slightly greater than the critical value, R-vc, meaning that at the onset of convection the base convectively unstable state can be viewed as marginally absolutely unstable. However, in several cases considered, the value of R-vt is significantly greater than the critical value, R-vc, implying that the base state is not marginally but essentially absolutely stable at the point of destabilization.