Append a row from newly added csv file to the old one in python

Question

I am trying to write a script which will be taking a particular column from a newly added to the folder csv file, and adding it as a row to an old csv file. So, the program should be executing this script whenever a new csv file is added to the folder resulting in one big csv file with many rows.

I have a program which appends csv files into one like this:

path = r'/Users/.../Desktop/files' # use your path

all_files = glob.glob(path + "/*.csv")

li = []

for filename in all_files:
    df = pd.read_csv(filename, index_col=None, header=0)
    li.append(df)

frame = pd.concat(li, axis=0, ignore_index=True)

frame.to_csv(r'/Users/.../Desktop/files/New.csv', index = False)

But it doesn't really do the full job, as what I want is this row from the newly added file (without a header):

to be appended into this type of file:

It is supposed to be happening whenever the new csv file is dropped to the folder, so maybe time.sleep() function should be used.

I really hope you can help.. I got totally lost when I tried to write the whole script.. :(

P.S. Sorry for the wordiness :/