I'm working on this now and Trophy Rush seems to be the way to go later in the game. While Crazy Orders & Classic will get you some moves, later on they seem to get you the same moves you already have, which stinks given the time those take to get through. And Smash Tour would occasionally get me some new moves even with a 15 turn game, but the last three I've tried have given me no new moves.
I've found a lot of success getting new ones in Trophy Rush though by using the characters I need moves for. Just last night I finished off five different characters in a row. Sure, sometimes you won't get a move, but you're only out 2 & a half minutes. Oftentimes you'll get a move for a character you're not even using either, in addition to a move for the one you're using. If you run low on money, do a Smash Tour run and go for the Coin reward battles. I still haven't stumbled on any homerun bat + first strike equipment for any single character, so I can only get as far as 18 or so matches, but even still you can rack up about 8000+ gold in that kind of run.
Also, I can't confirm if this is true or not in Trophy Rush, but I seemed to have a lot more luck getting new custom moves in for the character I was using when I wasn't using a custom. I went from using a few custom characters and would generally just get custom parts, sometimes a custom move, but using a default character I would frequently get custom moves. Could just be dumb luck or coincidence, but just throwing that out there.
Finally, doing this mode will also help get you towards that 50,000 trophy block goal, which takes a while.