Digitization of phase-modulated carrier signals with a commercially available analog-to-digital converter (ADC) is a common task in many communication and sensor applications. ADCs deliver phase-modulated digital carrier signals, which are numerically demodulated in order to extract the relevant information. However, the limited dynamic ranges of available ADCs limit the carrier-to-noise ratio of carrier signals after digitization. Correspondingly, the resolution of the demodulated digital signal is degraded. We demonstrate a sampling method with a simple demodulation scheme for phase-modulated signals with a small modulation index. Our new scheme overcomes the limitation due to digital noise defined by the ADC. Through simulations and experiments, we provide evidence that our method can improve the resolution of the demodulated digital signal significantly, when the carrier-to-noise ratio of phase-modulated signals is limited by digital noise. We employ our sampling and demodulation scheme to solve the problem of a possible degradation of measurement resolution after digital demodulation in heterodyne interferometers measuring small vibration amplitudes.