Extended Kaplan-Meier for survival curves from Cox PH model with time-varying covariates?

I am interested in using a proportional hazards model with external time-varying covariates for the purpose of predicting future event probabilities. Since extracting survival curves with time-varying data is not as simple as the time-independent case, I was wondering if the following procedure makes sense:

In KM it is assumed that this jumps only at event times, so if the event times are denoted as $t_1 < t_2 < ...$ then $$\hat S(t) = \exp(-\sum_{i | t \geq t_i} \hat h(t_i))$$ If there is an external time-varying covariate, say $x(t)$ with an estimated effect $\beta$. Then the "baseline" group corresponds to the hazard $h(t)$ and the "exposed" gorup to $h(t) \exp(x(t) \beta)$. You can see that, to estimate the $S(t)$ in the "exposed" group you only need the values at the event time points of this time-varying covariate.