  Based on the 13.7~yr Fermi-LAT data, Yeung et al. (2023) claimed detection of
two $\gamma$-ray sources (Src-NE and Src-NW) associated with the supernova
remnant (SNR) G298.6$-$0.0, and interpreted it as an old GeV SNR interacting
with molecular clouds (MCs). In this follow-up study, we refine the flux
measurements below 2~GeV with Fermi-LAT event types of better angular
reconstruction. Then, we report our cosmic-ray phenomenology in a hadronic
scenario, considering both the shell and MC regions of SNR G298.6$-$0.0. We
confirm that both the $\gamma$-ray spectra of Src-NE and Src-NW exhibit
spectral breaks at $1.50_{-0.50}^{+0.60}$~GeV and $0.68_{-0.11}^{+0.32}$~GeV,
respectively. Src-NW has a harder broadband photon index than Src-NE,
suggesting an appreciable difference between the physical separations of their
respective emission sites from SNR G298.6$-$0.0. The cosmic-ray spectrum
responsible for Src-NE starts with a minimum energy
$E_\mathrm{CR,min}=1.38_{-0.16}^{+0.47}$~GeV, and has a proton index
$\Gamma_\mathrm{CR}=2.57_{-0.21}^{+0.18}$ below the exponential cutoff energy
$E_\mathrm{CR,max}=240_{-150}^{+240}$~GeV. Accordingly, we argue that Src-NE is
dominated by the SNR shell, while only a minor portion of lower-energy emission
is contributed by the MCs interacting with the SNR. The cosmic-ray population
for Src-NW starts at a higher energy such that the $E_\mathrm{CR,min}$ ratio of
Src-NW to Src-NE is $\gtrsim$2. The high $E_\mathrm{CR,min}$, as well as the
high cosmic-ray energy density required ($\sim$26~eV~cm$^{-3}$), supports the
interpretation that Src-NW is predominantly the $\gamma$-ray emission from the
farther MCs being bombarded by protons that had earlier escaped from SNR
G298.6$-$0.0. By comparing the high-energy features of G298.6$-$0.0 with those
of analogical SNRs, especially SNR W28 and SNR W44, we further constrain the
age of SNR G298.6$-$0.0 to be 10--30~kyr.
