Excitation of antiferromagnetic HoFeO3 with a single 80 fs laser pulse triggers a first-order spin-reorientation phase transition. In the ultrafast kinetics of the transition one can distinguish the processes of impulsive excitation of spin precession, nucleation of the new domain and growth of the nuclei. The orientation of the spins in the nuclei is defined by the phase of the laser-induced coherent spin precession. The growth of the nuclei is further promoted by heating induced by the laser excitation. Hereby we demonstrate that in HoFeO3 coherent control of the spin precession allows an effective control of the route of the heat-induced first-order magnetic phase transition. The theoretical description of the excitation of the spin precession by linearly-polarized ultrashort laser pulses is developed with the sigma model. The analysis showed high sensitivity of the excited dynamics to the initial spin orientations with respect to the crystallographic axes of the material.