Calculating contrail levels was done in the good old days by The Met Office. They used to include a flight level called MINTRA or minimum trail altitude. This was primarily for the use of military pilots. Flying just below MINTRA would ensure that your position was not given away by contrails. The trail level can be calculated manually using a tephigram. The ascent curve for the current conditions is plotted until it reaches the line marked 'Mintra'. The level where the two lines cross is the 'exhaust trail' level. This used to be examined in the old style ATPL exams
A bit more background info:
'' To aid the forecasting of condensation trails emitted (or not) from high-flying aircraft, a line marking the critical temperatures (altitude dependent), above which trails are not possible, is marked on a tephigram (q.v.). The values are approximately -24degC at 1000 hPa (i.e. roughly sea-level), -39degC at 250 hPa (34000ft / 10.4 km) and about -45degC at 130 hPa (50000feet/15km). Using the MINTRA line (as it has come to be called - based on experiments by JK Bannon during World War II with the piston-engined Spitfire), a forecaster will mark two further lines on a tephigram: MINTRA minus 11degC (A) and MINTRA minus 14degC (B). If the ambient temperature (from the tephigram air temperature plot) lies between (A) and (B), then short, non-persistent trails are possible. If colder than (B), then long, persistent trails should be expected. However, some note should be paid to the relative humidity - high values will tip the balance to trailing (or longer/persistent trails.), even with air temperatures warmer than (A); ultra-low rh% will reduce the risk of condensation trails - the design of engines will have an effect as well. In broad terms, warm Tropical Maritime airmasses with a high but cold tropopause will result in a good deal of trailing, whilst cold, polar air-masses with a low, relatively warm tropopause will seldom give rise to significant aircraft trails.''