Well, you can't put it right above the first one, obviously, and if the tower is connected firmly to it's base your main problem is that the tower might look goofy since the first tap is centered. Maybe you can make it a 3-tap tower to look good - add two holes, one to either side of the original, and put them slightly higher or lower than the original (so they are closer to one another. Or blank out the center hole somehow and then add two holes just above or just below the original...
Is your tower stainless? If so, it will be hard to drill. If it is chrome, it will be really hard to penetrate the plating, but then it will drill ok, but you may snag onto some chrome plate and peel it off and ruin the tower (only cosmetically). Using strong tape over the area you plan to drill will minimize the risk of chrome plating peeling off, just drill through the tape and make sure there is tape all the way around the outside of the hole for a good 1/2".