Something i don't usually think about is that Haymitch and Effie were working together BEFORE katniss and peetas games.
Haymitch probably doesnt get close with tributes he "helps", he only helped katniss and peeta because they made an impression, they survived long enough, and they lectured him into it.
He was close with one person while mentoring, and thats effie.
in the first movie, she invites him to join her and peenis to dinner and said they were having haymitches favorite. which insinuates she KNOWS his favorite food.