You ask good questions. I will ask you good questions.
How moist is the pile? Did you water in between layers as you built the pile? When you squeeze a handful of the compost a few drops of water should come out. If it's a stream of water, too much.
What were the raw materials in the original pile?
3x3x3 is a good size. Larger is a better. As far as adding more material when turning... all depends. If there's something wrong with the pile, then maybe. If the pile heated up to 140 or more, I'll typically just turn it. But I also will add air to my pile simply by stirring up the top portion of the pile some. Once the temp reaches 140 and then starts dropping, I'll play in the pile with a pitchfork... stir it up some, typically that's all it needs to heat back up. Unless the pile has gotten too hot for too long. I know you aren't having that problem, but when you do get your pile heating up, temps above 150-160 for too long will dry out the center of the pile and turn it an ashy gray color. If I suspect it's been too hot for too long, compost thermometer helps a lot with that, then I'll just stir the pile some. Basically insert the pitchfork and lift. Let out some water vapor. If the top of the pile hasn't composted much, I might at the time also just turn the top into the middle without turning the entire pile. Only after the pile looks like it's composted a good amount, things turning dark brown, uniform in color, and the pile starts to cool, will I actually turn the pile. At that stage its the last time I'm going to mess with that pile, as I likely already have another pile in the works. I know that the pile might heat up once more, before cooling and then I'll let it cool compost for a while.
If my pile isn't heating or it's having problems composting, I rebuild the pile. First I try to determine why the pile isn't heating. For me usually not enough nitrogen. If the pile is slimy, smells like pond scum, mostly grass clippings then it probably has too much nitrogen and I'll add carbon materials, like leaves. I'll only use one new raw material and the pile that's having problems. For pile that isn't wet and slimy, I'll only use fresh grass clippings and the contents of the pile to rebuild. For a wet slimy pile, I'll only use leaves and the contents of the pile. I'll pick an empty bin, or new location for the pile and then start with a layer of new material followed by a layer of the material in the pile. If the pile is dry, I might water every couple of layers. Rinse, repeat. I do not water if the pile is too wet already.
So as the pile composts it reduces in size and you can keep adding material. The question then becomes when is the pile done? Well if you keep adding material to it, the pile isn't going to get done. I typically will build a pile and not have enough material to finish it in one go, so the pile will reduce as I wait on materials. I'll fill the bin to the top (mine are 4x4x4) and let is reduce. It'll typically reduce by half in a few days to a week. I'll top the pile off once or twice keeping an eye on the temp, and stirring on occasion. Maybe after adding material for a month, I'll say it's time to let this pile be, and start a fresh new pile. It could be less than a month, depending on how quickly you build the pile, but I typically will top off a couple times. For me, I have to move at least two piles every time I want to let a pile start to age, as I have a four bin system.
Hope this info helps and doesn't confuse you. I've suggested to others, this is how I compost. It's not the only way. You can just dump stuff in a pile and it's going to decompose. Most people's goal though is to provide a good environment to promote faster decomposition of the materials by all the beneficial organisms that are doing all the work.
One last note, you might want to take a look at the carbon to nitrogen ratios of materials. If you are using wood chips or something like that, you'll need a lot more nitrogen than most people think.
