read csv with filename column

import os
import glob
import csv

# import csv files from folder and change directory
path = r'file_path'
os.chdir(path)

# create list of all files in directory
allFiles = glob.glob(path + "/*.csv")

# read in each file, add filename column
with open('output.csv', 'w') as csvoutput:
   for i, fname in enumerate(allFiles):
       with open(fname,'r') as csvinput:
           if i != 0:
               csvinput.readline()
           writer = csv.writer(csvoutput, lineterminator='\n')
           reader = csv.reader(csvinput)
           print(os.path.basename(csvinput.name))

           all = []
           row = next(reader)
           row.append('FileName')
           all.append(row)

           for row in reader:
               row.append(os.path.basename(csvinput.name))
               all.append(row)
           writer.writerows(all)

File 1:
Col1    Col2    Col3
A       B       C
A       B       C
A       B       C
A       B       C

File 2:
Col14   Col5    Col6
D       E       F
D       E       F
D       E       F
D       E       F

Current output:
Col1    Col2    Col3    FileName
A       B       C       file_1.csv
A       B       C       file_1.csv
A       B       C       file_1.csv
A       B       C       file_1.csv
D       E       F       FileName  <---- script is not adding filename, but header
D       E       F       file_2.csv
D       E       F       file_2.csv
D       E       F       file_2.csv

Expected output:
Col1    Col2    Col3    FileName
A       B       C       file_1.csv
A       B       C       file_1.csv
A       B       C       file_1.csv
A       B       C       file_1.csv
D       E       F       file_2.csv
D       E       F       file_2.csv
D       E       F       file_2.csv
D       E       F       file_2.csv