Do a header check on the http:// version of your domain (http headers for Firefox is useful, or you can use an online tool instead). This way you can analyse how the redirect to https works. Make sure it's 301, not 302, as 302 seems to be the default for some platforms.
Do this check for some deeper pages too, not just your homepage.
Run something like Screaming Frog across your website, to make sure you've changed all internal linking to https. That means your XML sitemap too.
A few other things to check: Google Analytics, Google Search Console - make sure these are fetching your https site.
Go through all your social media channels, and change the links to https. Special attention to Google My Business, Google +.
Compile a list of all of your backlinks, and make the effort to change them to point to the https version of your domain. While this is time consuming, make sure your most authoritative links are changed. Not all are essential.