There is no specific date or month given in the text of the book. However, sometime in June is a reasonable assumption.
We know that the official establishment of the Super Suicide Society was on the evening of the day when Mr. Patch-Withers and his wife had hosted the term tea for the Upper Middle class of students. As a social event designed to assist the students and faculty in mingling and becoming acquainted outside the classroom,
three dozen of us stood tensely teetering our cups amid the wicker and leaves, trying hard not to sound as inane in our conversation with the four present Masters and their wives as they sounded to us.
it is logical to presume that this event occurred relatively early in the summer session.
After the tea ended, Gene and Finny needed some other activity.
"Let's go jump in the river," he (Finny) said under his breath as we went out of the sun porch...We need to clear our heads of that party," he said, "all that talk!"
They ended up, of course, literally jumping into the river from the river-side tree limb that became so central to their Society.
"We'll jump together to cement our partnership," he (Finny) said. "We'll form a suicide society, and the membership requirement is one jump out of this tree."
This decision was reached early in the summer session. All indications are that it was made sometime in June, 1942.