One possible layout

On desktop, opt for a header with 2 or 3 rows. There are various possible layouts, but below is a draft mockup of one of them:

 

Row 1

 

Row 2

 

Row 3

 

A variation on the layout above

Please ignore the colors though, it’s only a low-fidelity wireframe.
 
  • separated top bar for the free shipping and help link
  • tagline
  • no country
  • labels on the icons
  • added the opening hours
  • moved the phone number
  • highlighted Specials menu item
 
 

How tall should the header be?

I looked at 12 well-known brands, and the average is 21.5% (measured with a browser ‘real estate’ of 1247px tall). I’d recommend keeping it below 20%.

 

Hyperlink the elements in the promo bar

What I like in the promo/announcement bar in the image below (Cotton Mills), above the menu bar, is that the 3 unique selling ‘bullets’ are clickable. It allows shoppers to quickly get more information by clicking directly on those.