Lemon Lime and Bitters is great with any clear hard alcohol. Use 7up or Sprite with Vodka for example, add lemon and lime juices and some bitters. Very refreshing.
Off the top of my head, you should be good for gimlets, sours, and modified versions of the martini (pick up a bottle of dry vermouth for the real thing). The rum should pair nicely with the lime juice and the simple syrup.
I think this search engine might work for you to find other drinks.
Had 1800 Pomegranate & Regular Margaritas the other day, better than any other pre-mixed marg I can remember having. Still a little syrup heavy, alc light but it was expected.
In need of a Negroni