Travelling waves in woodpile chains are typically nanoptera, which are composed of a central solitary wave and exponentially small oscillations. These oscillations have been studied using exponential asymptotic methods, which typically require an explicit form for the leading-order behaviour. For many nonlinear systems, such as granular woodpile chains, it is not possible to calculate the leading-order solution explicitly. We show that accurate asymptotic approximations can be obtained using numerical approximation in place of the exact leading-order behaviour. We calculate the oscillation behaviour for Toda woodpile chains, and compare the results to exponential asymptotics based on tanh-fitting, Padé approximants, and the adaptive Antoulas-Anderson (AAA) method. The AAA method is shown to produce the most accurate predictions of the amplitude of the oscillations and the mass ratios for which the oscillations vanish. This method is then applied to study granular woodpile chains, including chains with Hertzian interactions -- this method is able to calculate behaviour that could not be accurately approximated in previous studies.