A few times I have looked at the possibility of switching from JPEG to another format for photo web sites and every time I have I've come to the conclusion that you can't really win.
There are three benefits that one could get from reducing the file size:
1. Reduced storage cost
2. Reduced bandwidth cost
3. Better user experience
In my models, storage cost matters a lot. You can't come out ahead here, however, if you still have to keep JPEG copies of all the images.
Benefits in terms of 2 are real.
Benefits in terms of 3 are hard to realize. Part of it is that adding any more parts to the system will cause problems for somebody somewhere. For instance, you can decompress a new image format with a Javascript polyfill, but is download+decompress really going to be faster for all users?
Another problem is that much of the source material is already overcompressed JPEG so simply recompressing it with another format doesn't lead to a really improved experience. When I've done my own trials, and when I've looked closely at other people's trials, I don't see a revolutionary improvement.
A scenario that I am interested in now is making desktop backgrounds from (often overcompressed) photos I find on the web. In these cases, JPEG artifacts look like hell when images are blown up, particularly when images have the sharp-cornered bokeh that you get when people take pictures with the kit lens. In that case I can accept a slow and expensive process to blow the image up and make a PNG, something like
https://www.mathworks.com/help/images/jpeg-image-deblocking-...
or
https://github.com/nagadomi/waifu2x
The other approach I imagine is some kind of maximum entropy approach that minimizes the blocking artifacts.