Unstructured Caps

I am a carpenter in the building & remodeling business. I am not in the embroidery or screen printing business. I'm not sure if my questions are welcomed here.
I have been wearing baseball caps, almost everyday, for 20+ years. I have always wanted a mesh back cap that was not stiff, or box like. I have finally found a cap I really like. The zkapz soft mesh trucker cap. They look and feel like a favorite old broken in cap, right out of the box. A friend saw me wearing a brand new one, and said "where did you dig up that relic, that's been around awhile, huh?" I like that.
I am very happy with the looks of my logo embroidered on the hat. The only thing that bothers me, is that my logo is not symmetrical. It has a big M on one side, which creates more structure on that side then the other. As the hats break in, the side without the big M, settles down more than the stiffer M side. The hats get folds in different places. They look unbalanced. The logo droops on one side. I am going to talk with the embroiderer about sewing in a symmetrical patch behind the logo. I decided to do some research first. A good hat is very important to a man who wears one everyday.
After looking around the internet, I noticed my logo seems longer than any others I have seen. I also noticed a lot of logos are taller. This got me thinking about the perfect shape and size for a logo, on an unstructured cap.
I was going to ask the embroiderer to sew in a rectangular patch behind the logo, on the next order (if that can be done), then I started thinking about a patch with a crown, or 1/2 circle top. Because I have also noticed these hats folding accross the top of the logo. It really depends on how the hat wearer wears the hat. I am attaching a picture of a hat I have been wearing, and a picture of one right out of the box. I would appreciate any advice or thoughts people might have. Thank you.