Not sure what you mean by 2nd time exit. The 2nd time exit was with my baggage, from the terminal 3 belt, after which I took a taxi out, ending the trip for good.

To go to T3 baggage claim, I had to go to the T3 arrival waiting area (where general public waits to receive passengers, you don't need any ticket/pass to go there), show my boarding pass and luggage tag to a Changi information counter staff, fill up a form with a lot of details (flight origin, number, name, passport number, pnr etc.) who issued me some kind of tag. With that I had to approach the CISCO certis desk (the counter manned by an extremely rude woman) who made me feel thoroughly guilty of my terrible oversight and making her do extra work, who arranged for frisking, took my passport, and gave me another tag. Then I had to collect my luggage, and go back to her to return the tag, and get the passport back. Then I exited via green channel of the customs.

One mistake was though, after exiting the plane, I went for quick washroom break, and hence lost the flow of passengers. After that, yes, it was a walk, but did not feel extra-ordinary, I did more walking at Jakarta CGK for example. But still, what you say, implies that terminals are seamlessly connected on the airside, without any manual checkpoint to prevent me from moving among them. While it may be great for transit passengers, seems an unnecessary potential trap for arriving passengers. A questionable design and flow control.

Will have to remember this at other major hubs at all. Flying to LHR next month. If something like this happens there, who knows if they will let me in again at all, or the luggage will be lost for good.