Airline safety: what happened between 1959 and 1962?
I believe the answer has something to do with the introduction of the passenger jet airliner, which first came into service in the late 1950s.  Note please that the original poster is asking for information on "commercial jets" whereas the statistics that others have offered have information on commercial planes in general, including prop planes.  I glanced at the statistics for 1959 presented above and there were only two passenger jets (including one KC-135, which you may discount if you like) that crashed.  

Compare this to the statistics presented for 1962.  There were six or seven crashes of Boeing and Douglas jet airliners.  

To me this means the differences were in all probablility not statistically significant.  We don't have enough "samples" to reach a valid conclusion.  

What you can see is a general explosion in the number of people flying.  Jets really introduced the public to flying.