You can bring the outlet tube up from the bottom on the outside of the coils if you want. It doesn't look as fancy as on the inside, but is easier to do. I built mine that way with 50 feet of 3/8 tubing. I like to very slowly stir the wort with my paddle in the center of the coils in order to keep it moving past the coils for better heat transfer - keeping the outlet tube outside the coil means it doesn't get in the way of the stirring. Keeping a slight down angle on the ends of the tubes, as in Nitsua's photo is a good idea - if the connection leaks, the tap water will flow away from the brewpot. Just be sure to make all the bends very gradual - my coil is about 8 1/2 inches diameter and the inlet and outlet bends are approximately as shown in Nitsua's photo, or maybe even less sharp than that. Be sure to use the spring tubing bender for the inlet/outlet bends (it's not needed for the coil if you wrap the tubing around a form and don't try to make too tight a coil). I found that my coil was hard to handle because it acted like big "slinky" toy, so I made a couple of big clips from some 6 gauge copper wire that I had on hand to hold the coil compressed to just the height of my brewpot, which left just a little space between the coils for wort to flow during stirring. I made the clips by bending the wire 180 degrees in the middle and slipping that over the bottom of the coil - the two pieces of wire run back up the sides of the coil, one inside, one outside. At the top, I bent them so that they hook together, but can be unhooked, kind of like a safety pin. Someday, if I get a bigger brewpot, I can make some larger clips, or maybe go ahead and solder some wire supports to the coils. I used compression fittings for my inlet and outlet connections - somewhere there is a thread with a parts list for this. It makes better connections, but is a bit pricey.
If you can salvage enough of the kinked tubing, you might want to make a pre-chiller coil. I have a 20 foot long prechiller made up from two 10 foot sections of scrap which I joined together with a 1 inch piece of vinyl tubing and hose clamps. I put the prechiller next to, but not in a bucket of ice water when I start the cooling. The tap water goes into the bottom of the prechiller, then from the top of the prechiller to the top of the wort chiller and from the bottom of the wort chiller back to the sink drain. After about 5 minutes of cooling, I set the prechiller into the bucket of ice water, then keep moving it up and down gently to get water moving across the coils. So I'm stirring the wort with my right hand and moving the prechiller up and down with my left - kind of like rubbing your stomach and patting the top of your head at the same time.
The prechiller doesn't actually speed up cooling all that much, but it lets me get the wort down to pitching temperature even when the tap water is warm in the summer. So far I've only used it in the winter with relatively cool tap water, but it gets four gallons from boiling to 70 degrees in ten minutes flat!