The winter in Minnesota is bleak for Dexter; he describes living in it as "a feeling of profound melancholy." Nonetheless, it offers him time and solitude to reflect on the direction he wants his life to take.
Counterintuitively, spring, for Dexter, is no better; for him, there is "something dismal" about it.
The fall, October and November, "filled him with hope" and an "ecstatic triumph," and energized him greatly.
Summer is recalled as a "fleeting brilliant impression" that fed his grandiose imagination, but it is also the season that drives him to pursue the dreams he indulged in during winter: to do more than observe and cater to the rich—to become one of them. The day he quits being a caddy on the summer golf course changes the direction of his life. The imagined life he concocts in his winter dreams becomes his reason for being, and his wealth begins with the humble purchase of a laundry.
It is clear from the title of this excellent short story by Fitzgerald that the seasons, or at least winter, will have a serious role in the action. It is in the opening paragraphs that we are given a description of how the seasons impact Dexter and which seasons are more important to him. Note how the changing of seasons changes Dexter:
Dexter knew that there was something dismal about this Northern spring, just as he knew there was something gorgeous about the fall. Fall made him clinch his hands and tremble and repeat idiotic sentences to himself, and make brisk abrupt gestures of command to imaginary audiences and armies. October filled him with hope with November raised to a sort of ecstatic triumph...
It is clear then that fall, with the hope of the approaching winter, makes Dexter feel more than he is - it makes him feel that he is an important man of value and commanding respect and loyalty from "imaginary audiences." It is no wonder then that because of this affinity he has with winter that he chooses to attach his "dreams" to this season - cold and elusive as they will turn out to be.