I've dealt with hotwire in some very creative ways over the years. So here's another idea.
I'm assuming your existing posts are wood.
I would put the no-climb clear down to the ground. It will keep your dogs in and everybody else's out. Unless they dig. (Whole 'nother topic.) Then I would buy a whole bunch of the nail-on foot long plastic insulators. Put one strand of electric at the top of the no-climb. Then, I would nail insulators sticking straight up on top of the posts and run another strand up there. That will increase the height of your fence by 12" with a hotwire. It is also possible to nail them at an angle so they stick out AND up. You could use the clip on insulators directly on the no-climb if you prefer the clip on kind. Then if needed, you can add another strand down lower.
Good luck! And congrats on getting your own place!