The double-polarization observable E was studied for the reaction γp→pω using the CEBAF Large Acceptance Spectrometer (CLAS) in Hall B at the Thomas Jefferson National Accelerator Facility and the longitudinally-polarized frozen-spin target (FROST). The observable was measured from the charged decay mode of the meson, ω→π+π−π0, using a circularly-polarized tagged-photon beam with energies ranging from the ω threshold at 1.1 to 2.3 GeV. A partial-wave analysis within the Bonn-Gatchina framework found dominant contributions from the 3/2+ partial wave near threshold, which is identified with the sub-threshold N(1720)3/2+ nucleon resonance. To describe the entire data set, which consisted of ωdifferential cross sections and a large variety of polarization observables, further contributions from other nucleon resonances were found to be necessary. With respect to non-resonant mechanisms, π exchange in the t-channel was found to remain small across the analyzed energy range, while pomeron t-channel exchange gradually grew from the reaction threshold to dominate all other contributions above W≈2 GeV.