Calculations of hyperfine splittings in ${\mathrm{H}}_{2}$ are needed for an accurate experimental determination of frequencies of rovibrational transitions and the dissociation energy in ${\mathrm{H}}_{2}$, which are used for experimental tests of quantum electrodynamics for molecules and for searches for new physics beyond the standard model. While the hyperfine structure of the ground electronic $X^{1}\mathrm{\ensuremath{\Sigma}}_{g}^{+}$ state in ${\mathrm{H}}_{2}$ has been studied in detail, there are no theoretical or experimental data regarding the excited electronic $^{1}\mathrm{\ensuremath{\Sigma}}_{g}^{+}$ states. Here, we report the first investigation of the hyperfine structure of rovibrational levels in the excited double-well $EF^{1}\mathrm{\ensuremath{\Sigma}}_{g}^{+}$ state in the hydrogen molecule. We provide hyperfine splittings and coupling constants for several low-lying rovibrational levels. The hyperfine splittings in the inner well are approximately 1 order of magnitude larger than those in the outer well and 2 times smaller than those in the ground electronic state.