When it snowed for the first time this winter, ...
When we had our first snowfall of the year, ...
in this winter ではなく this winter とすれば良いですよ。
ーWhen it snowed for the first time this winter, ...
次のようにも言えます。
ーWhen we had our first snowfall of the year, ...
例：
When we had our first snowfall of the year last week, I still hadn't changed to my winter tires, so I had to drive extra carefully.
「先週、今年初の雪が降った時、まだ冬のタイヤに変えていなかったので特に気をつけて運転しなければいけなかった」
ご参考まで！
カナダ 
