There is a way of shifting the problem to other curves: start with an unramified covering $\pi \colon D \to C$ that is Galois over $\bar{\mathbb Q}$ and compute the associated Selmer set $\operatorname{Sel}^\pi(C)$. The set is finite and computable, at least in principle; its elements $\xi$ correspond to twists $\pi_\xi \colon D_\xi \to C$ of $\pi$ such that $$C(K) = \bigcup_\xi \pi_\xi(D_\xi(K)),$$ which "reduces" the problem to that of determining the sets of rational points on the curves $D_\xi$ (whose genus is larger than that of $C$, so they tend to be more difficult to deal with). One can then hope that other methods can be applied to these curves. In some cases, for example, these curves map to other curves of lower genus over $K$ or over larger fields, and one might be able to find their $K$-points or the images of the $K$-points on $D_\xi$.
The problems with this approach in practice are:
- The computation of the Selmer set may be infeasible (it works reasonably well for 2-coverings of hyperelliptic curves, but not in many other cases);
- As already mentioned, the covering curves are "worse" than $C$, and one needs some luck to reduce the problem of determining $D_\xi(K)$ to something manageable (again chances are best in the hyperelliptic case);
- It is likely to be hard to get it to work for a family of curves like the $X^+_{\text{ns}}(N)$ that you mention in a comment above. The determination of $X_{\text{ns}}(13)({\mathbb Q})$ is a notorious open problem, even though the curve has genus only 3!