The classical-input quantum-output (cq) wiretap channel is a communication model involving a classical sender $X$, a legitimate quantum receiver $B$, and a quantum eavesdropper $E$. The goal of a private communication protocol that uses such a channel is for the sender $X$ to transmit a message in such a way that the legitimate receiver $B$ can decode it reliably, while the eavesdropper $E$ learns essentially nothing about which message was transmitted. The $\varepsilon $-one-shot private capacity of a cq wiretap channel is equal to the maximum number of bits that can be transmitted over the channel, such that the privacy error is no larger than $\varepsilon\in(0,1)$. The present paper provides a lower bound on the $\varepsilon$-one-shot private classical capacity, by exploiting the recently developed techniques of Anshu, Devabathini, Jain, and Warsi, called position-based coding and convex splitting. The lower bound is equal to a difference of the hypothesis testing mutual information between $X$ and $B$ and the "alternate" smooth max-information between $X$ and $E$. The one-shot lower bound then leads to a non-trivial lower bound on the second-order coding rate for private classical communication over a memoryless cq wiretap channel.