Get thee behind me Satan-Prime!
Neim, 9 bytes

&gt;ͻ:D+6S÷


Explanation:

&gt;         Increment input
 ͻ        Start infinite loop
  :        Previous prime
   D       Duplicate
    +6     Push 666
      S    Swap
          See if 666 is a substring of the top of the stack
        ÷  If true, break


Try it online!