Ultrafast magnetization dynamics in an epitaxially grown Co wedge-shaped layer sandwiched between Pt films has been studied by means of time-resolved magneto-optics. By changing either the external magnetic field or the thickness-dependent anisotropy field, we were able to tune the magnetization precession, as it depends on the effective magnetic field affecting the sample. A particularly interesting possibility of tuning occurs at the range of thicknesses near the spin reorientation transition (at nm), where the effective anisotropy field changes sign. To describe the measured dynamics we propose a model based on ultrafast thermal changes of the effective anisotropy field as a mechanism of triggering precession of the spins.