Linear mod one transformations are those maps of the unit interval given by $f_{\beta,\alpha}(x)=\beta x+\alpha$ (mod 1), with $\beta>1$ and $0\le\alpha<1$. The lap-counting function is $L_{\beta,\alpha}(z)=\sum_{n=1}^{\infty} L_{n}z^{n}$, where $L_{n}$ essentially counts the number of monotonic pieces of the $n$th iterate $f_{\beta,\alpha}^{n}$. Part I showed that the function $L_{\beta,\alpha}(z)$ is meromorphic in the unit disk $\vert z\vert <1$ and analytic in $\vert z\vert<1/\beta$, and part II showed that the singularities of $L_{\beta,\alpha}(z)$ on the circle $\vert z\vert=1/\beta$ are contained in the set $\{(1/\beta)\exp (2\pi il/N_{\beta,\alpha}):0\le l/N_{\beta,\alpha}\}$, where $N_{\beta,\alpha}$ is the period of the ergodic part of a Markov chain associated to $f_{\beta,\alpha}$. This paper proves that the set of singularities on $\vert z\vert=1/\beta$ is identical to the set $\{(1/\beta)\exp (2\pi il/N_{\beta,\alpha}):0\le l/N_{\beta,\alpha}\}$. Part II showed that $N_{\beta,\alpha}=1$ for $\beta> 2$, and this paper determines $N_{\beta,\alpha}$ in the remaining cases where $1<\beta\le 2$.