After grinding for all smash powers in smash run and then for all trophies in trophy rush and then for all custom moves in classic 7.9 I was only missing four hats:
Fancy Pirate Hat
Panda Hat
Floral Hat
Wedding Veil
I played a Smash or two with a Mii character so I didn't get a ton of character-specific rewards from target blast (though I don't think that helps at all really), then played target blast over and over with Villager, retrying if there was no custom item. I did this while watching One Piece, if not the most effective method it's at least the least boring. I got the first 3 hats the same day I started within only a few hours. Then the next day (sunday) I must've done that like 8 hours without the wedding veil showing up. Next time I played I did target blast for a while but got bored and proceeded to trophy rush as Donkey Kong. Got it in like 10 tries.
I really think it's completely random, though I thought the chance of getting new ones was much worse in classic, at least in 7.9.
I'm still not done, I still have 7 bonus effects on equipment to find. Those seemed pretty rare in all modes, I got 0 new ones while grinding custom moves and only 1 new one grinding hats. Since then I've only played Classic 9.0 once and got a new one right then, so maybe that's the way? Has anyone got them all?