Most adaptations are what I'd call first-order. Cats evolve better eyesight; redwoods evolve to grow taller; male cardinals evolve attractive bright feathers. All of these changes were selected for because they directly make the organisms and their offspring more likely to reproduce. But it seems like there is a fundamentally different minority of adaptations, which I'll call second-order. Second-order adaptations do not make organisms and their offspring more fit directly, but rather they make an organisms' lineage more quickly responsive to selective pressures. Sexual reproduction is certainly the best example. The primary evolutionary benefit of sex doesn't come from increasing immediate fitness. In fact, I'd argue that many organisms' fitness is reduced by the fact that they cannot reproduce without a partner. Sex evolved not because it confers fitness to individuals, but because it confers adaptability to lineages: sexually reproducing lineages are quicker to evolve to fill new niches and avoid new threats. This is why I'd call sex a second-order adaptation, because it seems to be a rare adaptation which improves the process of evolution itself rather than the evolving organisms.
Is my understanding correct here, that sex is a fundamentally different kind of adaptation from, say, prehensile tails? If so, are there any other documented examples of second-order adaptations? Is there more accepted terminology than "first-order" vs "second-order" for this distinction?
Also, can anyone think of a third-order adaptation or pressure? Does that even make sense?
Possibly related: this post. But to be clear, I'm not asking about epigenetics.