How many lattice paths in $\mathbb{Z}$ of length $n$ with steps in $\{-1,0,+1\}$ visit $m$ distinct points?
Notice that this is just the number of lattice paths $P$ such that $\max P - \min P + 1 = m$. If this helps, the generating function for the set of paths $P$ such that $\max P = k$ seems to be
$$(M(z) z)^k (M(z) z)^* M(z)$$
where
$$M(z) = \frac{2}{1 - z + \sqrt{(1 - 3 z) (1 + z)}}$$
is the generating function for the Motzkin numbers. The interpretation of this generating function is that one takes a left-leaning Motzkin path, moves right, takes a left-leaning Motzkin path, moves right, and so on $k$ times. Finally, one takes a left-leaning Motzkin path, moves left, takes a left-leaning Motzkin path, moves left, and so on an arbitrary number of times. Thus the final position can be anywhere to the left of the maximum.
According to The Range of a Simple Random Walk on $\mathbb{Z}$ by Bernhard Moser, the number of lattice paths in $\mathbb{Z}$ of length $n$ and range $d$ is
\begin{align} a_{n,d} &= (\mathbf{1}^\mathrm{T} \mathbf{Q}_d^n \mathbf{1} - \mathbf{1}^\mathrm{T} \mathbf{Q}_{d-1}^n \mathbf{1}) - (\mathbf{1}^\mathrm{T} \mathbf{Q}_{d-1}^n \mathbf{1} - \mathbf{1}^\mathrm{T} \mathbf{Q}_{d-2}^n \mathbf{1}) \\ &= \mathbf{1}^\mathrm{T} \mathbf{Q}_d^n \mathbf{1} - 2 \cdot \mathbf{1}^\mathrm{T} \mathbf{Q}_{d-1}^n \mathbf{1} + \mathbf{1}^\mathrm{T} \mathbf{Q}_{d-2}^n \mathbf{1} \end{align}
where $\mathbf{1} =(1, \dots, 1)^\mathrm{T}$ is a vector of ones and $\mathbf{Q}_d$ is the $d \times d$ band matrix $$ (\mathbf{Q}_d)_{ij} = [i - 1 \leq j \leq i + 1] $$
I include the main diagonal since I have steps $\{-1,0,+1\}$ rather than $\{-1, +1\}$ as in the paper. I've checked $a_{n,d}$ numerically and this expression seems correct.