Context. Despite the ever-increasing number of known exoplanets, no uncontested detections have been made of their satellites, known as exomoons. Aims. The quest to find exomoons is at the forefront of exoplanetary sciences. Certain space-born instruments are thought to be suitable for this purpose. We show the progress made with the CHaracterising ExOPlanets Satellite (CHEOPS) in this field using the HD 95338 planetary system. We present a novel methodology as an important step in the quest to find exomoons. Methods. We utilised ground-based spectroscopic data in combination with Gaia observations to obtain precise stellar parameters. These were then used as input in the analysis of the planetary transits observed by CHEOPS and the Transiting Exoplanet Survey Satellite (TESS). In addition, we searched for the signs of satellites primarily in the form of additional transits in the Hill sphere of the eccentric Neptune-sized planet HD 95338b in a sequential approach based on four CHEOPS visits. We also briefly explored the transit timing variations of the planet. Results. We present refined stellar and planetary parameters, narrowing down the uncertainty on the planet-to-star radius ratio by a factor of ten. We also pin down the ephemeris of HD 95338b. Using injection-and-retrieval tests, we show that a 5σ detection of an exomoon would be possible at RMoon = 0.8 R⊕ with the methodology presented here. Conclusions. We exclude the transit of an exomoon in the HD 95338 system with RMoon ≈0.6 R⊕ at the 1σ level. The algorithm used for finding the transit-like event can be used as a baseline for other similar targets, observed by CHEOPS or other missions.