Texan here, born and raised. I am 62, nearly 63 years old, and homecoming mums were a tradition even when I was in elementary school. When I was in high school, we couldn't wear them to school but only wore them to the football game, not the dance.
We saved our mums after the game, and hung them on our bulletin board (cork board) that we all had in our bedrooms. I saved all 4 of mine in the years I was in high school. Our mums were one large mums with very long streamers and charms like cowbells, footballs, year, high school initials like LHS, etc tied to the streamers. In glitter, We had our name and our date's (boyfriend's) name on one of the streamers, plus the year.
When my daughter was in high school in the 90's, they could also wear them to school. She was a varsity cheerleader and the cheerleaders also made mums to hang on each of the football players lockers. (I thought that was excessive.) Many of the girls wore more than one mum--one from their date, one from parents, and sometimes one from their best friend. When I was in high school, we only wore one mum, from our date, and only to the football game. The boys did not wear mums.
I have no idea when the tradition of mums started but I started elem. school in 1956 and the tradition was already going strong then.