Contrail cirrus introduce a short-lived but significant climate forcing that could be mitigated by small changes in aircraft cruising altitudes. This paper extends a recent study to evaluate the efficacy of several vertical flight diversion strategies to mitigate contrail climate forcing, and estimates impacts to air traffic management (ATM). We use six one-week periods of flight track data in the airspace above Japan (between May 2012 and March 2013), and simulate contrails using the contrail cirrus prediction model (CoCiP). Previous studies have predominantly optimised a diversion of every contrail-forming flight to minimise its formation or radiative forcing. However, our results show that these strategies produce a suboptimal outcome because most contrails have a short lifetime, and some have a cooling effect. Instead, a strategy that reroutes 15.3% of flights to avoid long-lived warming contrails, while allowing for cooling contrails, reduces the contrail energy forcing (EFcontrail) by 105% [91.8, 125%] with a total fuel penalty of 0.70% [0.66, 0.73%]. A minimum EFtotal strategy (contrails + CO2), diverting 20.1% of flights, reduces the EFcontrail by the same magnitude but also reduces the total fuel consumption by 0.40% [0.31, 0.47%]. For the diversion strategies explored, between 9% and 14% of diversions lead to a loss of separation standards between flights, demonstrating a modest scale of ATM impacts. These results show that small changes in flight altitudes are an opportunity for aviation to significantly and rapidly reduce its effect on the climate.