  CUPID-0, an array of Zn$^{82}$Se cryogenic calorimeters, was the first
medium-scale demonstrator of the scintillating bolometers technology. The first
project phase (March 2017 - December 2018) allowed the most stringent limit on
the neutrinoless double beta decay half-life of the isotope of interest,
$^{82}$Se, to be set. After a six months long detector upgrade, CUPID-0 began
its second and last phase (June 2019 - February 2020). In this letter, we
describe the search for neutrinoless double beta decay of $^{82}$Se with a
total exposure (phase I + II) of 8.82 kg$\times$yr of isotope. We set a limit
on the half-life of $^{82}$Se to the ground state of $^{82}$Kr of
T$^{0\nu}_{1/2}$($^{82}$Se)$>$ 4.6$\times \mathrm{10}^{24}$ yr (90\% credible
interval), corresponding to an effective Majorana neutrino mass m$_{\beta\beta}
<$ (263 -- 545) meV. We also set the most stringent lower limits on the
neutrinoless decays of $^{82}$Se to the 0$_1^+$, 2$_1^+$ and 2$_2^+$ excited
states of $^{82}$Kr, finding 1.8$\times$10$^{23}$ yr, 3.0$\times$10$^{23}$ yr,
3.2$\times$10$^{23}$ yr (90$\%$ credible interval) respectively.
