Once you are done saving the file, you could always split the file into file pieces or multiple files based on the number of lines.

split -l 1000 output_file

or even better just try

command | split -l 1000 -

This will split the output stream into files with each 1000 lines (default is 1000 lines without -l option).

The below command will give you additional flexibility to put or enforce a prefix to the filename that will be generated when the output is generated and splitted to store into the file.