To be honest, I've found it pretty random. When I was cycling for Genji there were certain villagers I introduced myself and my other character to, and they actually stayed longer. There was another who I just introduced to my mayor and they were pinging to move by the time I'd tt'd 4 days
What I do in my main town is make sure I speak to everybody before I statt tt'ing, then go day to day until I find somebody who wants to move (I never go past 4 days, I just change the clock back to the proper date and start over, it works best at 10 villagers too). So if I'm on the 11th but don't get a ping to move until the 13th, I'll just go straight back to the 11th, then check again for pings, go 4 days forward and check again, go back to the 11th. If I still haven't got a ping to move by this point (which happens occasionally, not very often though) I'll just go to the 12th, 13th, etc until I get one. Then back to the 11th. Does that make sense?