This paper investigates the spatio-temporal instability of the natural-convection boundary-layer flow adjacent to a vertical heated flat plate immersed in a thermally stratified ambient medium. The temperature on the plate surface is distributed linearly. By introducing a temperature gradient radio $a$ between the wall and the medium, we obtain a similarity solution which can describe in a smooth way the evolution between the states with isothermal and uniform-heat-flux boundary conditions. It is shown that the flow reversal in the basic flow vanishes when $a$ is larger than a critical value. A new absolute–convective instability transition of this flow is identified in the context of the coupled Orr–Sommerfeld and energy equations. Increasing $a$ decreases the domain of absolute instability, and when $a$ is large enough the absolute instability disappears. In particular, when $a\,{=}\,0$ (isothermal surface), the interval of absolute instability becomes narrower for fluids of larger Prandtl numbers, and the absolute instability does not occur for Prandtl numbers greater than 70; when $a\,{=}\,1$ (uniform-heat-flux surface) the instability remains convective in a wide Prandtl number range. Analysis of the Rayleigh equations for this problem reveals that the basic flows supporting this new instability transition have inviscid origin of convective instability. Based on the steep global mode theory, the effects of $a$ and Prandtl number on the global frequency are discussed as well.