We show that, thanks to a controlled use of strong laser fields in hyperfine spectroscopy of molecules, it is possible to record the crossover resonances between allowed ΔF = ΔJ lines and ΔF ≠ ΔJ lines which are very weak for high J lines. A demonstration experiment on the Q(45)A₂² line of the ν₃ band of SF₆ yields very accurate values of the spin-rotation and spin-vibration constants, c_a and A, whose ratio denotes the importance of purely vibrational effects.