Based on data samples collected with the BESIII detector at the BEPCII collider, the process e^{+}e^{-}→Σ^{+}Σ[over ¯]^{-} is studied at center-of-mass energies sqrt[s]=2.3960, 2.6454, and 2.9000 GeV. Using a fully differential angular description of the final state particles, both the relative magnitude and phase information of the Σ^{+} electromagnetic form factors in the timelike region are extracted. The relative phase between the electric and magnetic form factors is determined to be sinΔΦ=-0.67±0.29(stat)±0.18(syst) at sqrt[s]=2.3960 GeV, ΔΦ=55°±19°(stat)±14°(syst) at sqrt[s]=2.6454 GeV, and 78°±22°(stat)±9°(syst) at sqrt[s]=2.9000 GeV. For the first time, the phase of the hyperon electromagnetic form factors is explored in a wide range of four-momentum transfer. The evolution of the phase along with four-momentum transfer is an important input for understanding its asymptotic behavior and the dynamics of baryons.