First to Sail the Jet Stream Around the World
As your diagram shows, the jet stream actually has two circulating components: one is "around the earth" (which is why the transatlantic crossing is so much faster going East than going West), while the second is more like a vortex, with a circulating flow "around" the main direction. This circulating flow is vertical in two places: North of the jet stream it is downwards, and South of the jet stream it is upwards.

That makes for an "almost endless" lift - and you could in principle hover there. It would be very uncomfortable, as the winds in that region are quite strong and turbulent.

It reminds me of something I have seen near the beach: when the wind comes in from the sea, it is lifted up by the dunes; and people with hanggliders can jump off the dune, catch the wind, and literally stay in the air as long as they want (and as long as the wind stays up). If they "miss" the updraft when they first jump off the top of the dune, they are on the beach within five seconds; if they succeed, they fly for an hour or more. Right alongside the sea gulls who discovered the same trick many years ago. Example source:



I don't think you can prevent yourself from traveling around the globe when you do this - in other words, you would have to be content to travel in a mostly Easterly direction. I suspect (but do not know for sure) that the currents are too strong to allow you to traverse in the Westerly direction by ducking in and out of the Jet.