We develop a new technique for the path approximation of one-dimensional stochastic processes. Our results apply to the Brownian motion and to some families of stochastic differential equations whose distributions could be represented as a function of a time-changed Brownian motion (usually known as L and G-classes). We are interested in the epsilon-strong approximation. We propose an explicit and easy-to-implement procedure that jointly constructs, the sequences of exit times and corresponding exit positions of some well-chosen domains. In our main results, we prove the convergence of our scheme and how to control the number of steps, which depends on the covering of a fixed time interval by intervals of random sizes. The underlying idea of our analysis is to combine results on Brownian exit times from time-depending domains (one-dimensional heat balls) and classical renewal theory. Numerical examples and issues are also developed in order to complete the theoretical results.