Neko Atsume Question

I've been playing since February and have hit an impasse.
  1. I have everything. All the cats. All the goodies. All the remodels.
  2. Tubs is a greedy gus so I am also getting a lot of gifts.
  3. I just checked - I don't have all the mementos yet. So that's a goal at least.
  4. Plus, I'm trying to get a picture of Frosty with her hat.
    Does this only happen in the winter? Any suggestions on how to entice her out and don her hat?