  In gamma-ray bursts (GRBs), $\sim$ 100 - 1000 s after the prompt emission,
afterglow observations have consistently shown X-ray excesses detected in the
form of flares (XFs; in long GRBs) or extended emission (EEs; in short GRBs).
These observations are interpreted as emissions from jets launched by late
central engine activity. However, the characteristics of these late-time jets,
particularly the dissipation radius ($r_{\rm diss}$), Lorentz factor
($\Gamma$), and cosmic-ray loading factor ($\xi_p$), remain unknown despite
their importance. Here, in order to understand the properties of the late-time
jets with future multi-messenger observations, we estimate the detectability of
neutrinos associated with late-time emissions for a wide range of $r_{\rm
diss}$ and $\Gamma$, assuming $\xi_p=10$. We take into account external seed
photons from the cocoon around the jets, which can enhance the neutrino
production through photohadronic interaction in the jet dissipation region. Our
results are still consistent with the upper limit obtained by IceCube. Our
calculations indicate a promising prospect for neutrino detection with
IceCube-Gen2 through the stacking of $\sim 1000-2000$ events, for a wide range
of $r_{\rm diss}$ and $\Gamma$. We found that setting an optimal energy
threshold of 10 TeV can significantly reduce noise without negatively affecting
neutrino detection. Furthermore, even in the case of non-detection, we show
that meaningful constraints on the characteristics of the late-time jets can be
obtained.
