It is a long time that the SIMULATED ANNEALING (SA) procedure is introduced as a model-free optimization for solving NP-hard problems. Improvements from the standard SA in the recent decade mostly concentrate on combining its original algorithm with some heuristic methods. These modifications are rarely happened to the initial condition selection methods from which the ANNEALING schedules starts or the time schedule itself. There are several parameters in the process of ANNEALING, the adjustment of which affects the overall performance. This paper focuses on the importance of initial temperature and then proposes a lower temperature with low energy to speed up the process, using an auxiliary memory to buffer the best solution. Such an ANNEALING indeed starts from a "mushy state" rather than a quite liquid molten material. The mushy state characteristics depends on the problems that SA is being applied to solve for. In this paper, the Mushy State SIMULATED ANNEALING (MSSA) is fully developed and then applied to the popular Traveling Salesman Problem (TSP). The mushy state may be obtained by some simple methods like crossover elimination. A very fast version of a Wise Traveling Salesman, who starts from a randomly chosen city and seeks for the nearest one as the next, is also applied to initiate SA by a low-energy, low-temperature state. This fast method results in quite accurate solutions compared to the methods recently cited in the literature.