subj: etale covers of line bundles on an abelian variety

Is there an explicit decryption of finite etale covers of a line bundle $L$ on an abelian variety and its associated C-bundles $L^* = L \setminus A\times {0}$ (i.e. the C-bundle $L^*$ is $L$ without the zero section) ?

Pull-back along multiplication by $n$ map $n:A\rightarrow A$ gives a pull-back $n^*_A L' \rightarrow L$. A tensor power map $L'\mapsto L'^{\otimes n}$ gives rise to an etale map of C-bundles
$L^* \rightarrow L'*^{\otimes n}$, and thus if $L$ happens to be a tensor power, to an etale cover of L.

Can we obtain all etale covers of $L^*$ this way ?

subj: etale covers of line bundles on an abelian variety

Is there an explicit decryption of finite etale covers of a line bundle $L$ on an abelian variety and its associated C-bundles $L^o = L \setminus A\times {0}$ (i.e. the C-bundle $L^o$ is $L$ without the zero section) ?

Pull-back along multiplication by $n$ map $n:A\rightarrow A$ gives a pull-back $n^*_A L' \rightarrow L$. A tensor power map $L'\mapsto L'^{\otimes n}$ gives rise to an etale map of C*-bundles
$L^o \rightarrow L^{o\otimes n}$, and thus if $L$ happens to be a tensor power, to an etale cover of $L^o$.

Can we obtain all etale covers of $L^o$ this way ?