No, the remaining movement beyond the reset does not count toward the next Stress or Happiness Tracker movement.

When the Mood Tracker token reaches either the rightmost or leftmost space on the Mood Tracker, you immediately:

• Move the corresponding Happiness or Stress Tracker token 1 space to the right (unless it is already at the rightmost space, in which case it stays).
• Reset the Mood Tracker token to the center space.

Any remaining movement is not carried over after the reset. Each time the Mood Tracker reaches an end, the movement stops, the appropriate tracker is advanced, and the Mood Tracker is reset to the center. If a card effect would move the Mood Tracker multiple spaces and it reaches the end, only the movement up to the end is counted, and the rest is not applied after the reset.

This is supported by the example on page 12:

"Third, he scores 1 point for Onsen and needs to move his Mood Tracker token 2 spaces to the right on his Mood Track. After moving it 1 space, he has reached the rightmost space of the track, so he moves his Happiness Tracker token 1 space to the right and resets the Mood Tracker token to the center space. Then he moves the Mood Tracker token 1 space to the right."

This shows that after the reset, any remaining movement is applied starting from the center, not carried over from the previous movement. Each time the end is reached, the process repeats: move the tracker, reset, and continue with any remaining movement.

Source: page 11, page 12.

Source: Base Rules (pp. 11, 12, 15, 2, 7)