Finding a cell in an arrangement of simplices
If I am understanding your problem correctly, this is the problem of computing the face containing a given point in an arrangement of line segments. There is a randomized algorithm running in expected time $O(n\alpha(n)\log n)$, where $n$ is the number of line segments, and $\alpha(n)$ is the inverse Ackermann function. The algorithm computes the boundary of the face, and from that you can list the grid points if you want using a sweep algorithm, but the time to do that will of course depend on the number of grid points inside the face.