In the Shemoneh Esrei, the line "who makes the wind blow and the rain descend" is added to the Second Blessing from Shemini Atzeres. The request to "give dew and rain for a blessing" is not added to the Ninth Blessing until a variable time a few weeks later. Both then run until Pesach.
Have any other qualities, beyond the physical one of its possibly raining, ever been ascribed to this period between Shemini Atzeres and the time when rain is positively required?