I think Jack Frost is essentially winter personified (elf, fairy, man figure), but culturally we do tend to associate him with late autumn because that's when we get the first frosts. From my childhood I seem to remember he was referred to at these times more than at the dead of winter.

On waking up on the first frosty morning people are likely to say, "Ooh, Jack Frost is on the way." or "Jack Frost has visited."

So for me, the way we speak (and when we speak) about JF - we use him to acknowledge that autumn is changing over to winter / that winter is on the way.