This paper is a continuation of our study of the carbon monoxide vibration-rotation bands in late-type stars. The Δυ = 1 fundamental transitions of CO near 4.6 μm can be used as probes for the thermal structure of the upper atmospheres of cool stars, where observations using different spectral diagnostics recently have led to contradicting results: the chromospheric temperature rise, deduced from UV/visual line observations, cannot be reconciled with temperature profiles inferred from infrared molecular line observations. A possible solution to this dilemma is the "thermal bifurcation" scenario, where cool and warm areas co-exist at a given altitude in the upper atmosphere. The cool areas, controlled by radiative equilibrium dominated by the CO Δυ = 1 bands, are responsible for the observed, strong molecular absorption spectra; whereas the hot non-radiatively heated, "chromospheric" areas are observed predominantly in high-temperature UV/visual atomic lines. In the two-component model any observed spectrum is an average over the inhomogeneous stellar surface, and need not be representative of either component. The parameters of the empirical models can depend rather crucially on the spatial averaging properties of the observed species. In this investigation we focus on the cool, molecular atmospheric component. We apply a non-LTE spectrum synthesis code for stellar atmospheres between spectral types F5 and K5 in order to establish the CO fundamental bands as a diagnostic tool. Spectra computed with varying stellar parameters reveal the sensitivity of the CO spectra to these parameters. The strong lines in CO Δυ = 1 spectra are affected by elemental abundances and surface gravity but cannot be used easily to derive these properties. On the other hand, CO line intensities respond strongly to the thermal structure of the upper atmosphere and thus are sensitive temperature probes, even if the excitation of the bands deviates from local thermodynamic equilibrium. We examine the impact of non-LTE effects on CO spectra quantitatively, including errors introduced by uncertainties in the crucial cross sections for vibrational excitation of CO through atomic hydrogen. Our study demonstrates that the CO Δυ = 1 bands are valid probes for the temperature structure over a broad range of physical conditions and thus can contribute fundamentally to the study of the outer atmospheres of stars of late spectral type.