Actually, have you tried some of the other “sipping rums” recommended in this thread? I think a good rum will easily match a high-quality whiskey or tequila.
I’m surprised no one has mentioned Myers’s Rum. It seems to be behind every bar I have been to. Is it any good?
My ex liked Pyrat Rum XO Reserve and said it was the only rum she’d ever had that she could sip straight up. But that was before we moved to St. Kitts, where they make Brinley Gold vanilla rum. We brought a lot of that back to the states with us, and everyone’s loved it. You can find it all over the island in St. Kitts, but in the US it’s currently only avaialble in New Jersey, New York, and Pennsylvania. Fair warning, the taste of their vanilla rum has gone downhill a bit since they started making coffee and mango rum as well.
Also fair warning, I don’t drink, so what the hell do I know?
Inigo Montoya, if you’ve ever tasted New Orleans rum, you’d know it is the kind of stuff one would want to be paid to drink!
Really really bad stuff. 
My friends and I drink Gosling’s Black Seal rum, which makes the best mojitos.
For a mojito, mix rum with guarapo (fresh sugar cane juice). Crush fresh mint leaves and lime wedges with a mortar and pestle, put the lime and mint in a glass full of crushed ice (must be crushed!), and pour your rum/guarapo mixture over it. Stir well, and enjoy, amigos. One of the most refreshing drinks I’ve ever had, sweet and cool and flavored by the lime and mint, perfect for hot days. James Bond drank a mojito in Cuba in Die Another Day, the last movie, so I think that helped the classic Cuban cocktail gain new popularity.
Of course, living in Miami, we have plenty of access to Cuban restaurants and cafes where you can just order guarapo, and watch them feed a fresh stalk of green sugar cane into a huge juice-extracting machine, to get a cup to go.
… I thought I’d throw out Prichard’s Fine Rum made in Kelso TN. It’s relatively new on the market and the guy who makes it has won some awards I believe, it’s made with table grade molasses instead of blackstrap. Small distillery so you might have trouble finding it.