We performed a first-principles study of the structural, vibrational, electronic, and magnetic properties of ${\mathrm{NaMnF}}_{3}$ under applied isotropic pressure. We found that ${\mathrm{NaMnF}}_{3}$ undergoes a reconstructive phase transition at 8 GPa from the $Pnma$ distorted perovskite structure toward the $Cmcm$ postperovskite structure. This is confirmed by a sudden change of the Mn-F-Mn bondings where the crystal goes from corner-shared octahedra in the $Pnma$ phase to edge-shared octahedra in the $Cmcm$ phase. The magnetic ordering also changes from a $G$-type antiferromagnetic ordering in the $Pnma$ phase to a $C$-type antiferromagnetic ordering in the $Cmcm$ phase. Interestingly, we found that the high-spin $d$-orbital filling is kept at the phase transition which has never been observed in the known magnetic postperovskite structures. We also found a highly noncollinear magnetic ordering in the $Cmcm$ postperovskite phase that drives a large ferromagnetic canting of the spins. We discuss the validity of these results with respect to the $U$ and $J$ parameters of the $GGA+U$ exchange-correlation functional used in our study and conclude that large-spin canting is a promising property of the postperovskite fluoride compounds.