Developing simple ways to control spin states in spintronic devices is a crucial step towards increasing their functionality. MnTe is a room-temperature antiferromagnet with promising spintronic properties, including for thermospintronics and magnon-based devices. Here, we show that, in MnTe, less than 1% Li is sufficient to produce a dramatic spin reorientation as observed by neutron diffraction. The behavior of the 0001 magnetic Bragg peak reveals a significant reorientation of the Mn spins from planar in the pure material to almost completely axial with minimal Li doping. Temperature dependence of the magnetic peaks in Li-doped samples indicates that axial spins shift back to planar suddenly upon approaching the Néel temperature (TN=307 K). Density functional theory calculations support the idea that a shift in the Fermi level caused by doping is responsible for switching the material between two competing magnetic ground states. These results pave the way for developing easy switching of magnetic states in functional materials such as spintronic devices and topological insulators.