Arduino SD Card open file modes append / overwrite
You only need to open the file with FILE_WRITE and use file.seek(EOF) to go to de end of the file. After that you can write whatever you want that will be appended to the end of the file.

File outputFile = SD.open(LOG_FILE, FILE_WRITE);
outputFile.seek(EOF);
outputFile.println("Appended to the EOF");