MySQL - Move data between partitions aka re-partition
You need to split each partition into two partitions?  Something like this (please verify the syntax):

ALTER TABLE table1  REORGANIZE PARTITION p2015half1 INTO
    partition `p0` values less than ('2015-10-01'),
    partition `p0` values less than ('2016-01-01') ;


That is, take any number of partitions (1 in this case), copy the data into any number of new partitions (2 in this case).

But I need to ask "why"?  Would you show us SHOW CREATE TABLE and explain why you chose to use PARTITIONing.  I ask this question because everyone seems to have the impression that partitioning is beneficial.  Often it provides no benefit, especially not in performance.

The next question has to do with "billions of rows".  To do anything with that much data takes a long time -- simply because of I/O speed.  What kinds of queries are you doing on the table?  Do you have summary tables?  Do you purge "old" data?