The Traveling Salesman Problem (TSP) is one of the most well-known NP-hard optimization problems. Following a recent trend of research which focuses on developing algorithms for special types of TSP instances, namely graphs of limited degree, in an attempt to reduce a part of the time and space complexity, we present a polynomial-space branching algorithm for the TSP in an n-vertex graph with degree at most 5, and show that it has a running time of O^*(2.3500ⁿ), which improves the previous best known time bound of O^*(2.4723ⁿ) given by the authors (the 12^th International Symposium on Operations Research and Its Application (ISORA 2015), pp.45-58, 2015). While the base of the exponent in the running time bound of our algorithm is greater than 2, it still outperforms Gurevich and Shelah's O^*(4ⁿ n^{log n}) polynomial-space exact algorithm for the TSP in general graphs (SIAM Journal of Computation, Vol.16, No.3, pp.486-502, 1987). In the analysis of the running time, we use the measure-and-conquer method, and we develop a set of branching rules which foster the analysis of the running time.