I have the Dream Hat +1 as well. I do not know if this is specifically what MUST be done, but this is what I did and it triggered the quest. After collecting all the trees, setting them up in my MH and getting the first hat I went and found every child on the Smilebringer's list (not just the ones I was given letters by, but all 10 choices when he asks you which kids the letters are from). This was somewhat easier to find kids being RNG with Widescan, and if requested I can scout out and give locations later when I have time. Meanwhile, the lists of the possible kids are for Windurst and San'Doria:
WINDURST
Windurst Woods: No kids at all
Windurst Walls: Chomomo, Komomo, Purakako
Windurst Waters: Hororo, Mimomo, Myumimi, Shanruru
Port Windurst: Arukoko, Paruru, Pichichi
SAN'DORIA
South Sandy: Authere, Blendare, Foletta, Leuveret
North Sandy: Maurinne, Taurette
Port Sandy: Artinien, Auvare, Brifalien, Miene
I do not have a list for Bastok, sorry ><. If anyone has that please post. Anyhoo, once I gave a present to all 10 of the kids in Windurst, I received a gift from under the Windurst Christmas Tree in my MH from my moogle. Opening the gift got a nice Santa Fireworks and my Dream Cap +1!!!!!
Hope this helps ^^
Andi