You have identified a key part of the narration - how the seasons mirror the tone and events of the novel. Well done! The summer season and mood at Devon is described just after Finny has managed to talk him and Gene out of getting into trouble because they missed a meal through jumping out of the tree. Gene comments:
That was the way the Masters tended to treat us that summer. They seemed to be modifying their usual attitude of floating, chronic disapproval. During the winter most of them regarded anything unexpected in a student with suspicion, seeming to feel that anything we said or did was potentially illegal. Now on these clear June days in New Hampshire they appeared to uncoil, they seemed to believe that we were with them about half the time, and only spent the other half trying to make fools of them. A streak of tolerance was detectable; Finny decided that they were beginning to show commendable signs of maturity.
Note how a metaphor is used to describe the teachers - like snakes, they have uncoiled in the summer due to the warmth and are more permissive and accepting of students. They are "mellowed" with the warmer whether and those "clear June days".
Significantly, it is Finny's fall (helped by Gene) that marks the end of summer and the Fall which leads to winter. This is significant because it marks the end of innocence in Gene and a time when both he and Finny have to face up to unpleasant realities and come to terms with their relationship and with the War itself. Chapter Six begins by stating "Peace had deserted Devon" and then continues to state that although the surroundings were not affected by Fall that much and there was still the appearance of summer, winter was truly on the way:
But all had been caught up, like the first fallen leaves, by a new and energetic wind.... this was [Devon's] one hundred and sixty-third Winter Session, and the forces reassembled for it scattered the easygoing summer spirit like so many fallen leaves.
This change in tone is caught up by the introductory service of the school, which represented a kind of clamp-down - a return to authoritarian rules and regimes. Gene himself relates this change to Finny's "accident":
Still it had come to an end, in the last long rays of daylight at the tree, when Phineas fell.
This change in mood is reflected again by Gene's comment on the students and how they have changed now in the Fall:
We had been an idiosyncratic, leaderless band in the summer, undirected except by the eccentric notions of Phineas. Now the official class leaders and politicians could be seen taking charge, assuming as a matter of course their control of these walks and fields which had belonged only to us.
Thus this change in mood is kind of a pathetic fallacy - the innocence and fun of summer is ended in a number of ways. Firstly through the act of Gene pushing Finny off the tree and his "accident". Secondly through the students continued maturing and awareness of the War, and lastly because many students were beginning to think about signing up instead of finishing their schooling. The bleak realities of their existence coincide with the change in season. This should help you examine the role of the change of seasons in the rest of the novel. Good luck!