It's easy to find where you rank for keywords that you want to measure.
But how do you find out what other keywords you rank for that you don't know about?
Webmasters Stack Exchange is a question and answer site for pro webmasters. It only takes a minute to sign up.Sign up to join this community
It is almost impossible to get an accurate list of current terms and potential terms. Google Webmaster Tools does give you some insight. You know that already. As well, Google Analytics will give you some too though they are often hidden behind the dreaded (not provided). You can ferret out these keywords. Here is one link: https://blog.kissmetrics.com/unlock-keyword-not-provided/ It can be involved but likely well worth the effort.
Most online tools are junk. Too many are slow and way behind the curve. However, if you have money you can spend, SEMRush does seem to be much faster and can tell you on a day-to-day basis which terms you rank for in the search engines. This may help. You can visit the site and get the top-ten at least for free. They do limit the page views each day. It is costly to sign-up, however, it really does seem to be the only one worth the price. Others seem to fall really short of expectations. Unfortunately.
Another thing you can do will not give you a list per se', but may help to get a good feel for what is important. You can always do a site:example.com search against your site. Google lists your page in order of importance if you do not also list search terms. You can see what these page are and extrapolate what Google finds most valuable for your site.
The last thing I will recommend is using a web analysis tool. Ironically, one of the very best is free and can use two methods of analyzing your site: a bug and reading your log file. Check out Piwik. These can often give you a good idea of what terms return users to your site. However, not all terms are available because Google somewhat hides search terms for HTTPS searches. I am not sure why Google does this. It seems that they can balance the concern with search user privacy and the need of the webmaster to know how to make their site better.