This stepping to the side for the next stitch instead of crossing again across the back of the fabric with a long stitch is called “laying” your threads. “Laid threads” are made with stitches taken in this side-stepping manner.
You can also start with your first vertical stitch just below the very center of the last horizontal line, and build your vertical stitches out from there, to ensure that the vertical stitches are centered over the horizontal.
In this case, since I’m working on even-weave fabric, I’ve already counted out the spacing, so I don’t need to start in the center.
Work the vertical laid threads the same way you worked the horizontal ones, stepping to the side for each stitch. The arrows map the direction of the stitches.
When you finish the second layer, you’ll have a nice boxy grid.