Appending output to a file?
There is no existing way to append data to an existing file. Only modes you have listed in the question.  

You will have to go for native PHP functions (a mode in fopen() or FILE_APPEND in file_put_contents()). 

Some pointers: 

Consider using hook_watchdog(). You can filter what you need to log to the file using the type key of the passed array. 

Don't forget that there is also a module, syslog, to log these watchdog entries to syslog. 

You can also use dd() function from devel module to easily log specific data to a temp file. 


  dd()
  
  Logs any variable to a file named “drupal_debug.txt” in the site’s
  temp directory. All output from this function is appended to the log
  file, making it easy to see how the contents of a variable change as
  you modify your code.
  
  If you’re using Mac OS X you can use the Logging Console to monitor
  the contents of the log file.
  
  If you’re using a flavor of Linux you can use the command “tail -f
  drupal_debug.txt” to watch the data being logged to the file.
  
  dd($data, $label = NULL)