Thermal disc winds occur in many contexts and may be particularly important to the secular evolution and dispersal of protoplanetary discs heated by high energy radiation from their central star. In this paper, we generalize previous models of self-similar thermal winds – which have self-consistent morphology and variation of flow variables – to the case of launch from an elevated base and to non-isothermal conditions. These solutions are well-reproduced by hydrodynamic simulations, in which, as in the case of isothermal winds launched from the midplane, we find winds launch at the maximum Mach number for which the streamline solutions extend to infinity without encountering a singularity. We explain this behaviour based on the fact that lower Mach number solutions do not fill the spatial domain. We also show that hydrodynamic simulations reflect the corresponding self-similar models across a range of conditions appropriate to photoevaporating protoplanetary discs, even when gravity, centrifugal forces, or changes in the density gradient mean the problem is not inherently scale free. Of all the parameters varied, the elevation of the wind base affected the launch velocity and flow morphology most strongly, with temperature gradients causing only minor differences. We explore how launching from an elevated base affects Ne II line profiles from winds, finding it increases (reduces) the full width at half maximum (FWHM) of the line at low (high) inclination to the line of sight compared with models launched from the disc midplane and thus weakens the dependence of the FWHM on inclination.