Simulations of contrail-to-cirrus transition were performed with an LES model. In Part 1 the impact of relative humidity, temperature and vertical wind shear was explored in a detailed parametric study. Here, we study atmospheric parameters like stratification and depth of the supersaturated layer and processes which may affect the contrail evolution. We consider contrails in various radiation scenarios herein defined by the season, time of day and the presence of lower-level cloudiness which controls the radiance incident on the contrail layer. Under suitable conditions, controlled by the radiation scenario and stratification, radiative heating lifts the contrail-cirrus and prolongs its lifetime. The potential of contrail-driven secondary nucleation is investigated. We consider homogeneous nucleation and heterogeneous nucleation of preactivated soot cores released from sublimated contrail ice crystals. In our model the contrail dynamics triggered by radiative heating does not suffice to force homogeneous freezing of ambient liquid aerosol particles. Furthermore, our model results suggest that heterogeneous nucleation of preactivated soot cores is unimportant. Contrail evolution is not controlled by the depth of the supersaturated layer as long as it exceeds roughly 500 m. Deep fallstreaks however need thicker layers. A variation of the initial ice crystal number is effective during the whole evolution of a contrail. A cut of the soot particle emission by two orders of magnitude can reduce the contrail timescale by one hour and the optical thickness by a factor of 5. Hence future engines with lower soot particle emissions could potentially lead to a reduction of the climate impact of aviation.