High resolution Fourier-transform spectroscopy data of term values in the spin-orbit (SO) coupled first excited A(1)Σ(+) and b(3)Π states in KCs were obtained from (4)(1)Σ(+) → A(1)Σ(+) - b(3)Π, A(1)Σ(+) - b(3)Π → X(1)Σ(+), and (1)(3)Δ1→b(3)Π(0(±)) spectra of laser-induced fluorescence (LIF). About 3000 new rovibronic term values of the A(1)Σ(+) and b(3)Π(Ω) states were obtained with an uncertainty about 0.01 cm(-1) and added to the previously obtained 3439 term values in Kruzins et al. [Phys. Rev. A 81, 042509 (2010)] and 30 term values of the b(3)Π(0(+)) state levels below the A(1)Σ(+) state in Tamanis et al. [Phys. Rev. A 82, 032506 (2010)]. The data field was extended considerably, going down to vibrational level v(b) = 0 and up in energy to 13,814 cm(-1), as compared to previously achieved v(b) = 14 and E = 13,250 cm(-1). Overall 6431 e-symmetry term values of (39)K(133)Cs were included in 4 × 4 coupled-channel deperturbation analysis. The analytical Morse-Long-Range (MLR) function yielded empirical diabatic potentials for the A(1)Σ(+) and b(3)Π(0(+)) states while the morphing of the SO ab initio points [J. T. Kim et al., J. Mol. Spectrosc. 256, 57 (2009)] provided the empirical diagonal and off-diagonal SO functions. Overall 98.5% of the fitted term values were reproduced with a rms (root mean square) uncertainty of 0.004 cm(-1). The reliability of the model is proved by a good agreement of predicted and measured term values of the (41)K(133)Cs isotopologue, as well as of measured and calculated intensities of (4)(1)Σ(+) → A(1)Σ(+) - b(3)Π LIF progressions. Direct-potential-fit of low-lying v(b) levels of the b(3)Π(0(-)) component yielded the MLR potential which represents the 204 f-symmetry experimental term values with a rms uncertainty of 0.002 cm(-1). The Ω-doubling of the b(3)Π0 sub-state demonstrates a pronounced vb-dependent increase.