If you really want to multiclass it, you've got a couple things to think about. You have a couple of full casting levels, so going with something that would have fun with that works well. Pure casters will be saddened by having known spells lag 1-2 levels behind, but partial casters like paladins could have a lot of fun with it.
If you are planning on using cutting words a lot, look for a build that doesn't compete for your reaction. For instance if you were playing a polearm master or sentinel paladin you would have a lot of competition for it. If you plan on using other bardic inspiration as well you want a build that does not reliable need it's bonus action.
Cutting Words gets a heck of a lot stronger at 5th when inspiration dice both grow from d6 to d8 and refreshing on a short rest instead of a long. If you're aiming at Cutting Words as a defining ability, that's really the sweet spot. Plus this way you don't exit the class without grabbing the ASI at 4th which would have been a big hit. And you get 3rd level bard spells which is pretty cool.
Though talking about when to leave, Lore 6 gives Magical Secrets: 2 extra spells known from ANY list. (And all bards get Countercharm at 6th as well.)
Of course, at this point you might just realize that what adds well with bard would be - more bard. Really, they are pretty sweet to play.