I can give you my method~
First, I set a "starting date." This is the Date I will always TT back to either after a successful move or I tell a mover to sit their butt down and stay. Let's say my "Start date" is July 5th.
I'll keep TTing forward day by day until getting a mover. In order to spot the mover, I do a few things.
-I talk to each villager around 5-10 times. I want to get rumor dialogue. If they give me a village rumor (Peppy asks for gossip, Normal says they want to see rumors before they spread them, Jocks ask you to spread world record, etc.) I can safely assume there is no mover. To be safe though, I usually wait until I get 2 villager rumors. Sometimes the mover will fake you out, so always get the village rumor more than once (Rumors regarding you do not count).
-I also walk in front of all my villagers to see if they ping. If I get a ping for a move, one of two things happens.
a) If the mover is someone I want gone, I tell them to move.
b) If the mover is someone I want to stay, I make them stay.
If I allow the move, I TT to the animal's Box date and say goodbye, then TT back to my starting date. After this, I TT forward 4 days. So I go back to the 5th, load up, then save and quit and hit to the 9th.
On the 9th, you wanna check EVERY VILLAGER. Get at least 2 rumors and allow everyone a shot to ping before you deem it safe to TT forward. If you're paranoid, you can TT forward and make a new character to survey your town. If nobody is in boxes, reset and load your mayor. If someone is in boxes, reset, TT back to the previous day, and ping the mover.
If I don't allow the move, I stop the move and TT back to the 5th, then to the 9th.
Usually it works pretty well. The game usually needs a day or two to give you a new mover, so TTing 4 days usually guarantees somebody. However, you CAN get a mover on the "starting date." From my experience, it's rare, but that's why you want to be very careful and ping everybody after TTing from your starting date (You can also run around and see if you have a mover on the starting date, but I find due to how often it doesn't happen that it's generally not worthwhile. Usually I'll get rumors and pings frequently after TTing forward). If nobody pings on the day I land on, I TT 1 day at a time until I get a mover.
There are numerous cycling guides on the forum, but I've found this method works well for me. Once I fall into the pattern, 9 times out of 10 I have a mover every other reload. It's rare that I need to TT farther, but if I do, I just go 1 day at a time to be safe.
I've never lost a villager due to accidental TT. However, I've also used this method for ages now, and I'm used to it. I recommend taking it slow and moving day by day, and following the guides in order to get a feel before developing a method you feel works better for you.