Regardless of whether you have a new or existing site, you should have a system to ensure that your site is implementing HTTPS correctly. Installing a certificate is just the start of the process. You must make sure that different aspects of your site are referencing HTTPS correctly.
This includes the links in your pages, handling links to your site, and your analytics and search engine profiles.
Even if your site uses HTTPS, the HTTP portion of any page that includes HTTP content can be read or modified by attackers. When an HTTPS page has HTTP content, it's called mixed content. You might think the page is secure, but it isn't because of the mixed content.
When a page has mixed content, browsers have visual queues to alert the user to the insecure state. You cannot rely on seeing https in the address bar—look for the green lock....