Gilmour's G6 is indeed pretty versatile for rock work, and is a great tuning for someone coming from "underarm" guitar as it shares some common intervals. It gives a minor triad on top, a major triad in the middle, a big four-note power chord on the bottom, and leads feel pretty familiar to a guitarist.
As mentioned, Jerry Byrd's C Diatonic tuning is pretty good for melodic work, and it also has several of the same chord voicings from the C6 tuning built into it. However, it's a bit limited on a six-string instrument. It comes into its own on a seven-string as Jerry used it, or an eight-string with an added low C.
William Leavitt's tuning (the Berklee tuning mentioned above) is pretty versatile from a harmonic perspective, and is very capable for jazz. The diminished seventh chord on the bottom is particularly handy, as it contains every chord tone of a 7b9 chord (which occur often in jazz) except for the root. The chordal possibilities are great in this tuning, although it can involve a fair amount of bar movement to play busy melodies.
B11 (B or C#, D#, F#, A, C, E low to high) offers some interesting possibilities. One of the keys to this tuning is that on any given fret the four treble strings give you a sixth chord, and from there if you move down two frets, the four bass strings give you a ninth chord built on the same root. Useful for getting from a I to a IV chord by way of I6-I7(or 9)-IV.
On an eight-string instrument, an E13 tuning can be a useful one, particularly with a G# as the high string, although on six-string it might be too limited. Pretty sure a lot of really old country hits were done with this tuning.
Finally, you can definitely still get plenty of mileage out of C6. The only issue is that, like you said, everything sounds Hawaiian until you either figure out what you're doing or you join a Hawaiian band. Just have to be mindful of which grips you use. Another thing to keep in mind is that with the right string gauges, from C6 you're a quick and easy retune away from high A6, C6/A7, D9, B11, and E7 (this last one is really cool for blues, and with some careful behind-the-bar bends it does a pretty mean country pedal steel imitation too). If I were going to only have one six-string steel, that's the route I would go for versatility.