Fiddle is correct, the 10th animal must be from the campsite or someone else's town, either spotass, streetpass, or WiFi. c:

So the day right after an animal leaves is when the fence will show up for a new animal?  Puddles left my town on the 12th.  Yesterday there were no new fences (The 13th).  So, would I just make a new character every time before playing to look for the fence?

That is correct, you should have gotten a fence plot today, but I've seen it take longer than usuall sometimes in my spare town. Just make a new character everyday atleast 2-3 times to be absolutely sure no one will move in that day. Tomorrow is almost absolutely certain to bring a new animal to your town.

Puddles was your 9th right?

I guess it is 1-2 days.  I have Elise moving in now.  I used the trick and found her plot.  I am wondering if I reset will I get a new villager or will she just move her house somewhere else?

And yes, Puddles was my 9th.

You might possibly get a new villager when you reset as the plot of land isn't for certain.

elyocpal that depends, is Elise an animal you got from a trade/campsite? If yes, then she will change where her house plot goes, but you'll never get a different animal. c:

Oh I see, thanks for clearing that up with me Snappy.  I got her to move in a fairly nice spot so I decided not to test my luck.  I had another question though..

If I'm TTing to get a villager to move out for a trade and I'm concerned where a new one will move in, how will I do the new character trick along with TTing?  I tried to go on a main character, set the time to before 6am, make a new, delete the new, make another new in the "new" day, but plots didn't show up for 3 days so I just went and made a new character.  Do they not show up at 6am?  Is it around the afternoon/random time?  Since I can't set the time with a new character, because I can't save, I wasn't sure how to get around this.

The day I FINALLY got someone camping in my campsite someone completely random moved in as my 10th villager.  They were from a friends town, but I didn't even talk to this villager while I was there.  
PLUS it was Ricky, who took the liberty to move in over a path.  

Uh oh. My question was answered.

I'm still a bit unclear about the new character -> New Spot/Villager.

One of my villagers are moving out and now I should expect a fill in by the game as it's not the 10th villager.
When the day comes for the fence plot you make a new character to see where the plot is going to be and what villager. If you don't like either you can create a new character till they match up on what you want and then create a new home, effectivley sealing in that character and plot till you start up your Mayor file and then the villager's there for good.

But say you manage to get an ultra popular/one of your dream villager in the worst possible spot with the game's intent on filling in the 9th villager spot. Is there any guarantee that the same villager will be wanting to move in if you make a new character or is it a random selection each time?

Example: You start a new character, find the new gate plot w/ say Stitches moving in but it's right in line of a major pathway or he managed to take out your persimmon trees in moving. If you create a new character will Stitches be gone and a new villager will be in the new plot of land?

From my experience, you will eventually find the same villager plot again. I've tried this trick and I managed to get Simon's plot three times. Twice it was in the same location and once it was in a completely different location.

I did this trick back in WW to get a lot of favorite villagers, and now that someone moved out of my NL town, I'm thinking of trying it once again... I'm hoping it works out. ;w; (The villager that moved was my only Uchi, so I'm planning to search for another; maybe Phoebe. :'D)