Originally Posted by stut

It's absolutely the quickest way to transfer at Green Park. Not only is there less distance to cover, most of it is covered for you, mechanically.

The TfL-suggested route may be quicker, despite the two changes. Piccadilly to District at Hammersmith is a cross-platform change (if it's quiet, I'd suggest Barons Court instead, as it's a rather lovely station), and District to Jubilee at Westminster can be done on an escalator or lift without much walking.