  We study $\gamma$-ray emission following $^{239}\mathrm{Pu}(n,\mathrm{f})$
over an incident neutron energy range of $2 < E_i < 40$ MeV. We present the
first experimental evidence for positive correlations between the total angular
momentum generated in fission and the excitation energy of the compound nucleus
prior to fission. The $\gamma$-ray multiplicity increases linearly with
incident energy below the 2\textsuperscript{nd}-chance fission threshold with a
slope of $0.085 \pm 0.010$ MeV$^{-1}$. This linear trend appears to hold for
the average excitation energy of the compound nucleus between $9 < \langle E_x
\rangle < 19$ MeV. Most of the multiplicity increase comes from an enhancement
around a $\gamma$-ray energy of 0.7 MeV, which we interpret as stretched
quadrupole $\gamma$ rays that indicate an increase in total fission-fragment
angular momentum with excitation energy.
