Fronts propagating into an unstable background state are an important class of solutions to the cubic complex Ginzburg-Landau equation. Applications of such solutions include the Taylor-Couette system in the presence of through flow, and chemical systems such as the Belousov-Zhabotinskii reaction. Plane waves are the typical behaviour behind such fronts. However, when the relevant plane wave solution is unstable, it occurs only as a spatiotemporal transient, before breaking up into turbulence. Previous studies have suggested that the band of plane waves immediately behind the front will grow continually through time. We show that this is in fact a transient phenomenon, and that in the longer term there is a fixed-width band of plane waves. Moreover, we show that the phenomenon occurs for a wide range of parameter values, on both sides of the Benjamin-Feir-Newell and absolute instability curves. We present a method for accurately calculating the parameter dependence of the width of the plane wave band, facilitating future experimental verification in real systems.