Recently, we have proposed a new free boundary problem representing the bread baking process in a hot oven. Unknown functions in this problem are the position of the evaporation front, the temperature field and the water content. For solving this problem we observed two difficulties that the growth rate of the free boundary depends on the water content and the boundary condition for the water content contains the temperature. In this paper, by improving the regularity of solutions, we overcome these difficulties and establish existence of a solution locally in time and its uniqueness. Moreover, under some sign conditions for initial data, we derive a result on the maximal interval of existence to solutions.