In this paper we present some extensions of recent noncentral moderate deviation results in the literature. In the first part we generalize the results in \cite{BeghinMacciSPL2022} by considering a general Lévy process $\{S(t):t\geq 0\}$ instead of a compound Poisson process. In the second part we assume that $\{S(t):t\geq 0\}$ has bounded variation and is not a subordinator; thus $\{S(t):t\geq 0\}$ can be seen as the difference of two independent non-null subordinators. In this way we generalize the results in \cite{LeeMacci} for Skellam processes.