You are correct that the 3 tests reported at the bottom of your display are for the overall model.
Usual practice is to report the Wald test values for individual coefficients. Those are easy to get, as all you need is the covariance matrix of the coefficient estimates from the final model. They also can be combined easily into a test on multiple coefficients, for example combining all levels of a multi-level categorical predictor. The Wald test assumes, however, a symmetric shape of the likelihood profile with respect to changes in each predictor around its estimate.
With small sample sizes, a likelihood-ratio test is the gold standard. To evaluate a specific predictor of interest, you can do such a test on models with and without that predictor. The related profile-likelihood confidence intervals (CI) can also be more reliable than the Wald CI, as they make no assumptions on profile shape. Instead, they directly evaluate the likelihood as a function of predictor value (effectively refitting the model for each of a set of predictor values). Getting those CI can, however, be computationally intensive so they aren't so often calculated or reported.
Score tests on individual predictors are possible, as explained on this page that covers all 3 tests. They are done starting with a model that omits the predictor of interest, however, so there is no practical advantage over a likelihood ratio test, and it provides results that are less reliable. Can't say that I've ever seen individual-coefficient score tests reported for a multiple-regression Cox model.