In a recent paper, Dobronravov et al. (“On the length of shortest strings accepted by two-way finite automata”, DLT 2019) prove that the shortest string in a language recognized by an n-state two-way finite automaton (2DFA) can be at least symbols long, improved to in their latest contribution. The lower bound was obtained using “direction-determinate” 2DFA, which always remember their direction of motion at the last step, and used an alphabet of size In this paper, the method of Dobronravov et al. is extended to a new, more general class: the semi-direction-determinate 2DFA. This yields n-state 2DFA with shortest strings of length Furthermore, the construction is adapted to use a fixed alphabet, resulting in shortest strings of length It is also shown that an n-state semi-direction-determinate 2DFA can be transformed to a one-way NFA with states.