In Bulgaria with an invalid Schengen Visa. Have flight back home from Czech Republic, what should I do?

I have a friend who is in Europe. He went there for a conference that takes place in Czech Republic and another event that takes place in Bulgaria. He flew into Helsinki from Singapore and took another flight to Bulgaria.

Between switching flights from Helsinki to Bulgaria the people at immigration gave him the exit stamp. He thought he had a multiple entry visa but as it turns out he had only a single entry visa.

He has flights from Czech Republic to Singapore(home) booked for the 28th of July. Bulgaria allowed him to enter even though his Schengen Visa had the exit stamp.

I suggested that he should go around Europe and visit countries like Bulgaria that allow people with valid Schengen Visas. Take a flight to Prague on the day he flies out to Singapore and not exit the Airport. He is flying by Finair. Can he somehow make the flight out of a European Country(let's say Bulgaria) a connecting flight and not need a Visa and fly out of Czech Republic using the flight booked earlier? Obviously without exiting the Airport.

I had even suggested him to visit embassies around Sofia but he says after contacting them some of them said that they won't be able to process the visa in Sofia and they can only do so in his home country or country of residence(Singapore). The Netherlands embassy suggested that they can proceed with the Application but will first have to cancel the existing valid Schengen visa.