That 6 day method sounds cool and may work better than mine, but I'll post mine in case you still want it~
-Choose a starting date (i.e January 5th). That is the date you always TT back to. It'll help you keep your head straight.
-From your start date, TT 1 day at a time until you get a mover. There are 2 reliable ways to find this mover:
a. Talk to your animals for rumors. If at least 2 animals give you a "generic villager rumor" you can safely assume a mover doesn't exist. A great guide to rumors exists
here. You want at least 2 rumors because your mover will sometimes give you a rumor, and you'll be unaware he/she is moving~
b. Just go for the ping. You can run in front of all animals first and if you lack the ping, get the rumors next~
-If you want this animal to move, let him. If you want him to stay, make him.
-If you want the animal to move, TT to their move-out date, load up, say goodbye if you wish, then Save and Quit out. TT back to your starting date, load, Save and quit.
-If you want the animal to stay, simply TT back to your start date, load the game, then Save and Quit out.
The reason you return to the start date is to that the game processes the move/move denial. This allows you to get a new mover, actually, as soon as the day you TT back, which I originally didn't realize could happen~ So this is why:
-After you Save and Quit out of your starting date, TT FOUR DAYS into the future. Check for rumors and Pings.
-More often than not, you'll have a mover on this day. If you don't, TT one day at a time. You often won't need to go further than a few days unless you're unlucky, which I have been, but usually by Day 4 it's the first day someone is able consider the move~
It's a very THOROUGH method, and it takes some time to perfect, but I've never had a TT accident and while it's a little slower, it's definitely safe and quicker than going one day at a time and such. It's great for letting your grass grow and flowers spawn as well. But the MAIN REASON I like it is because your animals don't "forget" you, which skips the dialogue and lets them ping~
- - - Post Merge - - -
The only real risk to my method is that because of TTing back, you can have a mover on your "back date". There's a small chance that when you TT 4 days forward, you can have a mover the next day. Be thorough. Until you're comfy with it, use a new villager to "scout" for boxed animals in order to avoid accidents until you find your rhythm~