Imagine a rotating skyhook with the majority of its length in the thermosphere, but whose hook side (as opposed to the counterweight side) has the ability to deploy wings or sails. Suppose that orbital and rotational parameters were chosen such that when the hook dips into the atmosphere, its translational velocity is similar to the average velocity of the local air currents - like a moving car's wheel rotating over the ground with static friction. Since wind currents vary, there would usually be a small (compared to orbital velocities) difference causing the hook side to experience winds. Now, obviously, we must be clever about which wind tool we choose and when we deploy it and how we trim it, but in principle, could there be enough available $\Delta V$ to counteract the loss in momentum from hurling payloads further into space?
While the question of a solar sail could change this calculus, I'm specifically asking about wind sails, kites, wings and other structures that would interact with the upper atmosphere.
As you can see, a boring old wing acting directly against the tether with no wind helping would have a significant forward force component until nearly verticality. At this point, the wings would be either trimmed to reduce drag or deflated and reeled in. Naively, a downward force on the body acts in the wrong direction, but trading altitude for speed could actually help if one were clever about the timing and ellipticity. This and similar maneuvers could be repeated as needed to regain orbital velocity, rotational velocity and altitude until the next launch.
Supposing the tether releases its payload at the top of its arc (when the payload's velocity has no vertical (earth radial) component), the tether would then be moving more slowly than the atmosphere by some amount, and there would be an expectation of tailwinds on hook re-entry. To take advantage of these, the tether could deploy a boring old sail in addition to the wing.