Yes, if you haven't tried rotating your handlebars upward, definitely try it. That's the easiest fix, if it works for you. Keep in mind that small changes can make big differences in how you feel on the bike, and you can't rotate the bars too much or it will become uncomfortable, maybe even impossible, to get your fingers around the brake levers.

As KnottedYet pointed out, the thing about smaller bars is that, while they don't bring the top bar closer to you, everything else ends up being closer and easier to reach: the brake hoods, brake levers, and the drops, if you use them. But again, I'd try the easier changes first: the rotation, and moving the position of the brakes up higher. Bike comfort is such a personal thing, and you never know what's going to help you until you try it.