Get the name of the 2nd non-blank column for each row

Question

I have the following pandas dataframe:

     A    B    C
0  1.0  NaN  2.0
1  NaN  1.0  4.0
2  7.0  1.0  2.0

I know I can get, for each row, the name of the first non-blank column with this script:

df['first'] = df.dropna(how='all').notna().idxmax(axis=1).astype('string') 

but how can I get the name of the second non blank column? This is the expected output:

     A    B    C first second
0  1.0  NaN  2.0     A      C
1  NaN  1.0  4.0     B      C
2  7.0  1.0  2.0     A      B

Thanks