Context. The recent IRLOS upgrade for VLT/MUSE narrow field mode (NFM) introduced a full-pupil mode to enhance sensitivity and sky coverage. This involved replacing the 2 × 2 Shack-Hartmann sensor with a single lens for full-aperture photon collection, which also enabled the engagement of the linearized focal-plane technique (LIFT) wavefront sensor instead. However, initial on-sky LIFT experiments have highlighted a complex point spread function (PSF) structure due to strong and polychromatic non-common path aberrations (NCPAs), complicating the accurate retrieval of tip-tilt and focus using LIFT. Aims. This study aims to conduct the first on-sky validation of LIFT on VLT/UT4, outline challenges encountered during the tests, and propose solutions for increasing the robustness of LIFT in on-sky operations. Methods. We developed a two-stage approach to focal-plane wavefront sensing, where tip-tilt and focus retrieval done with LIFT is preceded by the NCPA calibration step. The resulting NCPA estimate is subsequently used by LIFT. To perform the calibration, we proposed a method capable of retrieving the information about NCPAs directly from on-sky focal-plane PSFs. Results. We verified the efficacy of this approach in simulated and on-sky tests. Our results demonstrate that adopting the two-stage approach has led to a significant improvement in the accuracy of the defocus estimation performed by LIFT, even under challenging low-flux conditions. Conclusions. The efficacy of LIFT as a slow and truth focus sensor in practical scenarios has been demonstrated. However, integrating NCPA calibration with LIFT is essential to verifying its practical application in the real system. Additionally, the proposed calibration step can serve as an independent and minimally invasive approach to evaluate NCPA on-sky.