Well there isn’t much scope in PNG for improvements. Because it is lossless, the only way to improve it is to spend longer looking for repeated patterns in the picture. Presumably someone felt they wanted a middle point between slowest and fastest, but it may not make a difference in many cases. Depends on the image.
Photoshop's PNG export and the old Save for Web are somewhat limited in regards to optimization options. Preprocessing an image with various techniques may reduce the size of an image drastically, though. Refer to How To Optimize PNG – Smashing Magazine, for example (it is an old article, but the information is still correct - excepting that the current releases of Photoshop can export 8bit indexed PNG files with full transparency).
However, I suggest getting ColorQuantizer instead, since it will save you a tremendous amount of time, and will result in even smaller file sizes anyway in almost all cases.
Get it here (download the latest version): Color quantizer
Save your work as a full 24bit (+ 8bit alpha channel if required) quality PNG file, and open it in CQ. You'll find various settings to finely control the optimization process under the small colour palette icon. Dithering amount, gradient<->colours balance, threshold for rare colours, dithering method, colour space, and more can be controlled. Specific colours can be controlled as well during conversion.
Aside from that a quality brush allows for creating a quality mask on the fly to tell CQ which parts of an image should be preserved at a higher quality. And finally it is possible to reduce the number of colours to any number - not just 256.
If PNG optimization (best image quality and smallest file size) is important to you, Color Quantizer is a life-saver. No other PNG optimization tool on the market compares. If you are running a Mac, use WINE to run it - it's worth it. Best of all: it's free.
Btw, if you are saving PNG files to maintain a lossless quality archive of images, I suggest you look into lossless WebP, which creates smaller file sizes than PNG. Unfortunately Photoshop is still unable to natively open or export WebP images (crazy in this day and age!!!), so a plugin is required.