A simple-triangle graph is the intersection graph of triangles that are defined by a point on a horizontal line and an interval on another horizontal line. The time complexity of the recognition problem for simple-triangle graphs was a longstanding open problem, which was recently settled. This paper provides a new recognition algorithm for simple-triangle graphs to improve the time bound from $O(n^2 \overline{m})$ to $O(nm)$, where $n$, $m$, and $\overline{m}$ are the number of vertices, edges, and non-edges of the graph, respectively. The algorithm uses the vertex ordering characterization that a graph is a simple-triangle graph if and only if there is a linear ordering of the vertices containing both an alternating orientation of the graph and a transitive orientation of the complement of the graph. We also show, as a byproduct, that an alternating orientation can be obtained in $O(nm)$ time for cocomparability graphs, and it is NP-complete to decide whether a graph has an orientation that is alternating and acyclic.