This is a useful site for getting maps of the weather conditions at contrail height, in North America or Europe.
At the top, click "Maps", then select "GFS", then the region you want, e.g. "CONUS", then "Upper Air". You get this menu:
The left hand drop-down menu lets you choose the time of the weather model run. You can go back about a week.
The second one chooses the parameter. Relative Humidity and Temperature are the most useful for contrails.
The third one is the height. 300mb = about 30,000ft. 250mb is about 34,000ft. Not all parameters are available at all levels. (See millibars to altitude table here
The fourth one is the forecast time, e.g. in this screenshot, 3 hours on from 2014-07-31 06Z will be for 09Z, or 9am UTC on July 31.
A few caveats:
One, it's a model, so it is necessarily an approximation of the true complex weather patterns. As you step further forward, it will get less accurate. However, if you choose the "0h", i.e. starting point, for a time in the past then it should give a good picture.
Two, and related to this, it's a large-scale overview. Very local variations in weather conditions won't show up.
Three, not all contrail heights are shown. It might be very humid at 300mb but much less so at 250mb.
Four, don't forget the temperature. It can be really humid, but if the air is too warm then you won't get persistent trails.
Nevertheless, it is a very good tool for explaining, and even predicting, contrail formation.