The double-polarization observable $E$ was studied for the reaction $\gamma
p\to p\omega$ 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, $\omega\to\pi^+\pi^-\pi^0$,
using a circularly-polarized tagged-photon beam with energies ranging from the
$\omega$ 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 $\omega$ differential 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, $\pi$
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 $W \approx 2$ GeV.
