Tensor-train (TT) decomposition has garnered tremendous popularity for its efficiency in handling high-dimensional data arising in scientific and quantum computing as well as machine learning applications. It provides a compact representation for matrices and vectors with a Kronecker product-like low-rank structure and enables efficient matrix-vector operations in this compressed form. The vector scalar product is among such key operations, comprising a series of tensor contractions in a specific tensor network topology whose order significantly impacts the computational cost. In this work, we propose efficient algorithms for finding near-optimal contraction orderings for tensor networks representing scalar products in the TT format. We show that our algorithms outperform all existing contraction ordering methods for general tensor networks where the best existing method incurs up to 15% higher cost for \(x^Ty\) , twice the cost for \(x^TAy\) , and ten times higher cost for \(x^TABy\) scalar products where x, y and A, B are vectors and matrices expressed in the TT format, respectively.