I'm not sure I particularly understand Super Smash Bros. Melee's Adventure Mode being better than Super Smash Bros. Brawl's Adventure Mode since the latter is just the former taken to its logical conclusion. If a new Adventure Mode of the latter variety were to be outsourced, we could end up with a mode that would basically be Super Smash Bros.: The Beat 'em Up with a super neat story, and that's just cool.
If you're just looking for more short form storytelling though I do have a suggestion:
- Each character or set of related characters has their own Adventure Mode.
- More Adventure Modes unlock as you beat the ones originally available to you.
- As you go through them, the characters events begin to intermingle.
- The final story (or stories) bring everyone together and has the big bad final boss at the end.
So basically, a small SSBCU.
For a different kind of Adventure Mode, how about this: World of Light, but now it's a metroidvania. Well, not literally just the World of Light again, but you get the idea. Instead of items, progression in this mode is gated by character abilities. For example, you could have barriers that you need Mewtwo, Zelda, Sheik, Meta Knight, Greninja, or Palutena to teleport past, or maybe a Sudowoodo is blocking the way, and you need Mario's F.L.U.D.D. or Squirtle's Water Gun to get it to go away. You could also have some secret areas hidden by character specific tech like Snake's infinite recovery.
Ooh! You could even have a hint system where characters discuss how to proceed.