Well, I'm a great fan of clawhammer. I'm really not very good at it, but it's a style that adapts to the re-entrant tuned ukulele very well and can be used as a "backing strum" for a sound that's a little bit different from the "usual"
down-up-down-up that seems to be used, in it's various forms, for most ukulele backing accompaniment. With a little further effort, clawhammer can be used for melody as well as just backing, so can make for a nice instrumental "break" between verses without breaking the overall "flow" of the number being performed.
I'm not a great one for "pidgeon-holing" the various picking styles ... Travis picking, Carter picking, "Celtic folk style" etc. etc. all seem to be variations on one another.
If you really want a formalised challenge, try "classical guitar style". Takes a lot of dedicated concentration to start out with, but the net result can be worth it if you want to play complicated melodies ... and you can even use the techniques you've learnt for other styles of music
YMMV, but enjoy the journey