How to visit Jewel in Changi on layover without paying extra?

If just seeing the artificial waterfall is what you’re after, you can do that at no extra cost from inside the transit area without ever entering Singapore. The transit shuttle from B to E (see Kate’s map) passes through the Jewel at a nice height to see the waterfall and the surrounding open space; see these pictures I took from inside that transit shuttle in April 2019:

Left: View of the waterfall and interior of Jewel. Right: the tracks of the automated people mover passing through Jewel. Click on the pictures for high resolution.

My memory is not entirely fresh, it could have been that the shuttles briefly waited in front of the waterfall to provide best views but even if they didn’t they don’t travel that quickly.

Note that while there is only one arrival platform in each of the terminal wings, the bit through Jewel serves as a passing area to double the capacity of the shuttles so if you’re in the unlucky direction there’ll be another path of the shuttle between you and the water. Fret not, the shuttle is free and goes back and forth all day so just board again.

Also note that if my memory serves me correctly Kate’s map is not accurate; you want to be on the inner circle to get a view of the waterfall (i.e. the shuttle passes through the part of Jewel closer to terminal 1).

I will also point out although it is at best tangential to the other points in this answer that on the other leg of my flight with Singapore Airlines that April (Asia–SIN–Europe return) I had an 18-hour layover, entered Singapore visa-free and went sightseeing in the city without having to pay any additional fee to get back airside. At no point in that transit did I pass any booth that had anything about a departure tax mentioned so reading about it here is the first time I saw it.

You don't have to pay anything extra. If you are eligible for visa on arrival, or VFTF, then you can clear immigration and visit. It is right next to Terminal 1. Even if you have around 2 hour layover, you can comfortably visit.

There is an official tour also which can be availed free of charge. It will take longer and your layover needs to coincide with the tour timings. It also includes free admissions to otherwise chargeable areas.