You should be able to replace the cassette with a cassette for a road bike with the same number of sprockets. You will likely have a chance to choose one with a smaller sprocket for high (higher gear), and probably closer tooth counts for closer shifts. You may find the chain is too long, but this can be fixed by taking a link out. Good luck.
If my previous answer doesn't get you a high enough gear, you might have to change the front chainrings to have a higher gear. A typical big ring on a hybrid is about the size of the middle ring of a road triple.
I agree that making a change to the front chainrings might be the thing to do. It's a bit of a pain in the butt though.
Average guy, father of three, works out at LA Fitness where I focus on strength training with some cardio on the side.