Context. Solar jets are collimated plasma ejections along magnetic field lines observed in hot (extreme-ultraviolet (EUV) jets) and cool (chromospheric surges) temperature diagnostics. Their trigger mechanisms and the relationship between hot and cool jets are still not completely understood. Aims. We aim to investigate the generation of a sequence of active-region solar jets and their evolution from the photospheric to the coronal heights using multithermal observations from ground-based and space-borne instruments. Methods. Using the synergy of high-spatial-resolution and high-temporal-resolution observations by the Swedish 1-m Solar Telescope (SST), along with the Solar Dynamics Observatory (SDO), we analyzed a sequence of solar jets originating in a mixed-polarity region between the leading and following sunspots of an active region. We investigated the kinematics of these jets using the spectra from the SST observations. We used a non-force-free field (NFFF) extrapolation technique to derive the magnetic field topology of the active region. Results. A mixed-polarity region is formed over a long period (24 hours) with persistent magnetic flux emergence. This region has been observed as an arch filament system (AFS) in chromospheric SST observations. In this region, negative polarities surrounded by positive polarities create a fan surface with a null point at a height of 6 Mm detected in the NFFF extrapolation. SST observations in the Hβ spectral line reveal a large flux rope over the AFS moving from north to south, causing successive EUV and cool jets to move in the east–west direction and later towards the south along the long open loops. Conclusions. The high-resolution SST observations (0″.038 per pixel) resolve the dark area observed at the jet base and reveal the existence of an AFS with an extended cool jet, which may be the result of a peeling-like mechanism of the AFS. Based on the combined analysis of SST and AIA observations along with extrapolated magnetic topology, it is suggested that the magnetic reconnection site may move southward by approximately 20 Mm until it reaches a region where the open magnetic field lines are oriented north–south.