Originally Posted by CPH-Flyer

Whether you change at Ginza or at Kasumigaseki does not really matter, either will work. My phone app suggests Kasumigaseki, but I can't say with certainty which offers the shortest walk between tracks.

Shibuya is also a good part of the loop towards Shinjuku, but the taxi ride is then very short. The taxi ride from Tokyo station is between 5 and 6 km, so it will depend on traffic. On a good day, yes it might be faster.