It all really depends on which character I'm using. I will say though that I'm a big fan of the Mii Masks because they allow for more creativity and uniqueness, so I tend to alternate between using those or actually using the in-game items.
For my main town's mayor, my favorite piece of headwear would be the Red Pumpkin Head. It's a horror-themed town and Halloween is my favorite day of the year, so yeah. Aside from the standard orange one, I find the Red-Pumpkin Head looks the nicest and find that her outfits tend to look better with that one than with the other colored Pumpkin Heads.
When she doesn't have a big jack-o'-lantern on her head, other hats and accessories I like for her are the Cow Bone, Flamenco Hat, Gas Mask, Hockey Mask, Jester's Mask, Mummy Mask, Red Hood, Red-Horned Hat, and the Witch's Hat. Sometimes I feel like just going without anything big and either just don't use anything to show off the face and hair or use something like one of the hairpins.
I have a vampire character in the same town and she alternates between using a Mii mask during the night and using various headcoverings and umbrellas during the day so that she can somewhat safely travel outside during the daytime (because vampire mythology can be a bit loose, and because it would be a shame to not use the character for a majority of the time the stores are open). I'll often use the Red Hood in combination with the Huge Shades since her outfits usually have black and red, though sometimes I'll swap out the hood for the Floppy Hat or other big hat similar in design. I'll also use the Kaffiyeh or Diver's Hood if I want to use the Oval Shades. The point is really just to keep the sun exposure limited. Sometimes I'll use the Black Veil or the Stagehand Hat as well since they seem fitting. Maybe sometimes the Hair Bow Wig as a reference to the 1992 movie Bram Stoker's Dracula and the silly looking hair they gave Dracula in that movie.
There's another character who I plan to make who will get a lot of use out of the Coin Headpiece.
I have other characters in other towns as well. Not very many of them have many defining accessories or headwear, save for one of them being a clown who I use the Afro Cap, Afro Wig, Balloon Hat, Bun Wig, Jester's Cap, and Red Nose for. Aside from her and the other characters already listed, none of the other characters I've made really have well-defined accessories other than maybe glasses for those who use them.