When I went to work for Allrecipes International as their lead UX Designer, their website was not responsive for mobile and was also in need of a complete redesign. Because time was a critical factor and mobile browsing was growing exponentially, we did not have time to redesign a responsive solution from scratch. The first phase was to convert the exiting website, as is, to responsive. Then go back, one page at a time, and redesign and replace all its components with newer designs.
Below are examples of full-page redesigns as well as individual component redesigns.