For several shift-invariant spaces, there exists a real number \(a\in \mathbb {R}\) such that the set \(a+\mathbb {Z}\) is a complete interpolation set. In this paper, we characterize the complete interpolation property of the set \((a+\mathbb {N}_0)\cup (\alpha +a+\mathbb {N}^{-})\) for shift-invariant spaces using Toeplitz operators. Using this characterization, we determine all \(\alpha \) for which the sample set \(\mathbb {N}_0\cup \alpha +\mathbb {N}^{-}\) forms a complete interpolation set for transversal-invariant spaces. We introduce a new recurrence relation for exponential splines, examine the zeros of these splines, and explore the zero-free region of the doubly infinite Lerch zeta function. Consequently, we demonstrate that \(\left\langle \frac{m}{2}\right\rangle +\mathbb {N}_0\cup \alpha +\left\langle \frac{m}{2}\right\rangle +\mathbb {N}^{-}\) is a complete interpolation set for a shift-invariant spline space of order \(m\ge 2\) if and only if \(|\alpha |<\tfrac{1}{2}\) .
