Sometimes it is helpful to first reduce the number of colors (preferably to 256 if that doesn't cause too much banding, depending on the number of color shades used). Then png usually compresses a lot better. Png compresses badly when the image contains too many different colors.
> Png compresses badly when the image contains too many different colors.
Old trick to squeeze a few kb out of a png: Use Posterize filter from Photoshop, with very light settings. Basically it will just flatten the number of colors.
Nah, just use Photoshop's "Export -> Save for Web (Legacy)", then set the file to PNG-8.
Now you can mess around with the number of colors in the color table, customise the color palette selection algorithm, dithering algorithm and dither amount.