  Magnetic reconnection is invoked as an efficient particle accelerator in a
variety of astrophysical sources of non-thermal high-energy radiation. With
large-scale two-dimensional particle-in-cell simulations of relativistic
reconnection (i.e., with magnetization $\sigma\gg1$) in pair plasmas, we study
the long-term evolution of the power-law slope and high-energy cutoff of the
spectrum of accelerated particles. We find that the high-energy spectral cutoff
does not saturate at $\gamma_{\rm cut}\sim 4\sigma$, as claimed by earlier
studies, but it steadily grows with time as long as the reconnection process
stays active. At late times, the cutoff scales approximately as $\gamma_{\rm
cut}\propto \sqrt{t}$, regardless of the flow magnetization and initial
temperature. We show that the particles dominating the high-energy spectral
cutoff reside in plasmoids, and in particular in a strongly magnetised ring
around the plasmoid core. The growth of their energy is driven by the increase
in the local field strength, coupled with the conservation of the first
adiabatic invariant. We also find that the power-law slope of the spectrum
($p=-{\rm d}\log N/{\rm d}\log \gamma$) evolves with time. For
$\sigma\gtrsim10$, the spectrum is hard at early times ($p\lesssim 2$), but it
tends to asymptote to $p\sim 2$; the steepening of the power-law slope allows
the spectral cutoff to extend to higher and higher energies, without violating
the fixed energy budget of the system. Our results demonstrate that
relativistic reconnection is a viable candidate for accelerating the
high-energy particles emitting in relativistic astrophysical sources.
