Laser-induced fluorescence spectra of the c³Σ⁺(v_c,J_c=N_c)→a³Σ⁺(v_a,N_a=J_c±1) transitions excited from the ground X¹Σ⁺ state of ³⁹K¹³³Cs 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 a³Σ⁺ state for levels with v_a∈[0,27] and N_a∈[24,90] shows that the splitting monotonically increases with v_a. The spectroscopic study was supported by ab initio calculations of the magnetic hyperfine interaction in X¹Σ⁺ and a³Σ⁺ states. The discovered variation of the electronic matrix elements with the internuclear distance R is in a good agreement with the observed v_a-dependencies of the HFS. Overall set of available experimental data on the a³Σ⁺ state was used to improve the potential energy curve particularly near a bottom, providing the refined dissociation energy D_e=267.21(1) cm⁻¹. The ab initio HFS matrix elements, combined with the empirical X¹Σ⁺ and a³Σ⁺ 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.