Jinglefruit
The black-nosed Reindeer
Firstly, it requires said villager to be outside and secondly it requires them to be the first one to ping you. This doesn't always happen.
=.= Except that 5 days before they try to move, providing you spoke to them within the past week and they are awake, they are always outside and they almost* always ping first to tell you this. - I have already said this and I have cycled for long enough to know that this is the case.
*The almost is because if you are close friends with a villager occasionally they ping you first thing in the day to request a favour of you, this doesn't negate the person moving outs ping though and they are able to ping immediately after this.
Also, it's NEVER taken me 3 minutes. It varies between sometimes very quickly (ten mins) to an hour (due to said villager not being about or other villagers pinging me). Sometimes I found out via gossip from other villagers and then have to do the diving trick to force that specific villager to ping me. Now maybe you have a super reliable method that always works (which I'd love to hear)
I do have a reliable method. In fact I've practically told you in my last few posts seeing as it requires nothing more than loading the game each day and running in front of each villagers face to see if they ping, then on every 6th day I talk to each villager once. (which takes a little longer cause half the time 1 is hiding in the museum.) One 7 day cycle takes me 20-25 minutes and since I've done this with my cycle towns I have not had a single villager leave without pinging me. Between these cycle towns I've travelled over 4 years, day by day.