Laser-induced fluorescence spectra of the c 3 Σ + (vc, Jc = Nc) → a 3 Σ + (va, Na = Jc ± 1) transitions excited from the ground X 1 Σ + state of 39 K 133 Cs molecule were recorded with Fourier-transform spectrometer IFS125-HR (Bruker) at the highest achievable spectral resolution of 0.0063 cm −1 . Systematic study of the hyperfine structure (HFS) of the a 3 Σ + state for levels with va ∈ [0, 27] and Na ∈ [24, 90] shows that the splitting monotonically increases with va. The spectroscopic study was supported by ab initio calculations of the magnetic hyperfine interaction in X 1 Σ + and a 3 Σ + states. The discovered variation of the electronic matrix elements with the internuclear distance R is in a good agreement with the observed va-dependencies of the HFS. Overall set of available experimental data on the a 3 Σ + state was used to improve the potential energy curve particularly near a bottom, providing the refined dissociation energy De=267.21(1) cm −1 . The ab initio HFS matrix elements, combined with the empirical X 1 Σ + and a 3 Σ + PECs in the framework of the invented coupled-channel deperturbation model, reproduce the experimental term values of both ground states within 0.003 cm −1 accuracy up to their common dissociation limit.