I think you can move out your 10th villager, it's just rare. I can post the method I use to move out villagers, if you'd like, since it's pretty safe.
- - - Post Merge - - -
The mechanics of this are identical to travelling two days at a time. Each day in ACNL starts at 6 AM. When you go back to a time before 6 AM that day, you've time travelled back to the day before, which is the same as one day forward. When you wait for 6 AM again, that's the same as one day forward again, since a new day begins at that time. It sounds like it would work.
The part about changing the time to 5 or 6 PM is completely irrelevant. As long as it's an hour when all your villagers are awake (so you can talk to them), it doesn't matter what it is. I think the villagers who wake up latest are Uchis, at 11 AM, so any time after that but before their bedtimes should be fine.
Waiting for pings isn't a good method, though, because sometimes villagers don't immediately ping you. Talking to them makes things slightly safer since you can hear gossip about who's thinking about moving. I don't like methods that rely on moving gossip or pings, though.
I hope this helps with understanding the game mechanics