How late in the flu season is a flu shot still worthwhile?

I went to the doctor today and he suggested I get the flu shot still even though it’s almost January since I’ve been sick 3 times this year so far. I keep getting the common cold, so getting the flu shot will actually help prevent more occurrences in the next few months. But I must advise that most doctors do not give anymore flu shots since as you stated, it’s normally done in October. He said I could go to Walmart or Walgreens and they will probably still sell it.