You could easily fly Turkish Air with a stop in Turkey, then on to Tirana and then work your way north and east until you have seen the Balkan states, Romania and Bulgaria (most not in Schengen); then cross over into Hungary and start your tour through Western Europe; ending in the UK (also out of Schengen). I suspect that could be well planned to work. I know a company that might be able to do the planning for you.
I am in Europe now and running a bit tight on my Schengen time so I am using that as an excuse for a return trip to Romania in about 2 weeks. My second choice was to take a discount flight to Istanbul for a few nights. In the spring or summer the choice would be easy, the Balkan states for fishing.
The Schengen calculator you found might be great, I use this one: https://www.schengenvisainfo.com/visa-calculator/
So, Schengen includes:
Austria, Belgium, Croatia, Czechia, Denmark, Estonia, Finland, France,
Germany, Greece, Hungary, Iceland, Italy, Latvia, Liechtenstein, Lithuania,
Luxembourg, Malta, Netherlands, Norway, Poland, Portugal,
Slovakia, Slovenia, Spain, Sweden, and Switzerland.
Which means you need to duck into one of these countries to cover yourself (or get out of Europe all together)
Albania, Armenia, Azerbaijan, Bosnia and Herzegovina, Bulgaria
Cyprus, Georgia, Ireland, Kazakhstan, Kosovo, Moldova, Montenegro
North Macedonia, Romania, Serbia, Turkey, Ukraine, United Kingdom (UK)