An increasing headwind will have to
keep increasing to infinite speed to do much good for more than a few seconds. Pretty rapidly, the inertia of the aircraft is overcome, and it is just drifting backwards with the headwind. No more relative headwind at the wing's leading edge, and a decrease in ground speed. Either of which will make achieving "long distances" - ummm -
challenging.
This is more or less just a restating of the "turn into the wind" question - unless an aircraft is fixed in position (tied down on the ground), the external wind has no effect on
airspeed - which is what produces lift. Except for very brief moments (gusty turbulence or shear) where a gust may act faster than inertia can be overcome.
You fly into the increasing headwind along the edge of the jet stream, and maybe gain a few seconds of extra lift and a bit of altitude. And you fly a little further into the shear and gain a few more feet, but soon you will be in the core of the jetstream, with no more shear, just a steady headwind. What do you do? If you fly out towards the edge of the jet, you now have a
decreasing headwind/shear and will lose altitude again. TANSTAAFL ("There ain't no such thing as a free lunch.")