Modify existing sticker design and maybe the logo

Project Description:
I have made a design for a sticker; the design is attached. The design is a sticker for an Italian handmade shaving soap. The design dimensions are fixed, I can't change those.

I'm not happy with this design, it doesn't express the company and product's real meaning. I also have a logo and it's not present in the design, I want it to appear in the sticker.

I'm happy with the logo as it is now, but feel free to change it if you think you can improve it. I'm open to suggestions. What I don't like much for the logo is the font used.

What I want the sticker to contain:
. Company logo
. Important: Express the following concepts: natural, shaving soap, highest possibile quality , Italian, handmade
. Ingredients, usage, and company info (these are mandatory texts, pretty much as is now, but it can be moved somewhere else, with a different color and font).

I need a design genius, someone who knows how to communicate.
I want an Illustrator vector design for the sticker and maybe a review of the logo.