Previously, we generalized the Lagrange polynomial interpolation at Chebyshev nodes and studied the Lagrange polynomial interpolation at a special class of sets of nodes. This special class includes some well-known sets of nodes, such as zeros of the Chebyshev polynomials of first and second kinds, Chebyshev extrema, and equidistant nodes. In this paper, we view our previous work from a different perspective and further generalize and study the Lagrange polynomial interpolation at a larger class of sets of nodes. In particular, the set of optimal nodes is included in this extended class.