The five-letter route
Trace GILLS through these cells: 1 at row 6, column 1; 2 at row 6, column 2; 3 at row 5, column 2; 4 at row 5, column 3; 5 at row 4, column 3. Reading the letters in that order produces G, I, L, L, S without leaving the listed path.
The movement pattern is right, up, right, and up. It contains three turns and four straight segments across a span of three rows and three columns. The route begins with three open orthogonal neighbors and finishes at a cell with one, giving you a useful endpoint check. After these five cells are assigned, 33 open cells remain.