The magnetic penetration depth (λ) in thin superconducting films is usually measured by the mutual inductance technique. The accuracy of this method has been limited by uncertainties in the geometry of the solenoids and in the film position and thickness, by parasitic coupling between the coils, etc. Here, we present several improvements in the apparatus and the method. To ensure the precise thickness of the superconducting layer, we engineer the films at atomic level using atomic-layer-by-layer molecular beam epitaxy. In this way, we also eliminate secondary-phase precipitates, grain boundaries, and pinholes that are common with other deposition methods and that artificially increase the field transmission and thus the apparent λ. For better reproducibility, the thermal stability of our closed-cycle cryocooler used to control the temperature of the mutual inductance measurement has been significantly improved by inserting a custom-built thermal conductivity damper. Next, to minimize the uncertainties in the geometry, we fused a pair of small yet precisely wound coils into a single sapphire block machined to a high precision. The sample is spring-loaded to exactly the same position with respect to the solenoids. Altogether, we can measure the absolute value of λ with the accuracy better than ±1%.