A natural extension of a right-continuous integer-valued random walk is one which can jump to the right by one or two units. First passage times above a given fixed level then admit - on each of the two events, which correspond to overshoot zero and one, separately - a tractable probability generating function. Some applications are considered.
"A note on the times of first passage for `nearly right-continuous' random walks." Electron. Commun. Probab. 19 1 - 7, 2014. https://doi.org/10.1214/ECP.v19-3735