I use the felted hangers. They don't take up much room.
I've seen tips where you hang one hanger on the bar of the one next to it, so you will have two coats hanging in one space. There's a visual in this video: https://www.youtube.com/watch?v=xaXKoga6Z40
Space saving hangers designed for particular clothing are available for purchase. The examples below are for pants (first and second pictures) and skirts or shorts.( third and fourth pictures)
Use all the same shape of hangers as they will hang closer together. Use all long coats first then the shorter ones. It may be possible to hang two rows of shorter coats with a long wire extender from rail to run under the existing rail.. Best wishes.
Work vertically! Try to fit shelves or hooks way up high for infrequently used items.
Here is a whole post I wrote about organizing and I have closet info in there too along with another post just about linen closets. I hope this helps! https://lifeonsummerhill.com/organizing-101/