Since you are using a grid and know which direction the user is proceeding there is nothing constraining you from adapting the prior answer and using a different fields of view depending on the direction.

For example you could extended the field to include the corners when you travel in cardinal directions and shrink it down two squares on each end in your diagonal cases so that each uncovers 9 squares.

Another alternative depending on how your lighting works would be to use a better circle approximation with aliasing, to partially reveal some of the squares.

Since you are using a grid and know which direction the user is proceeding there is nothing constraining you from adapting the prior answer and using a different fields of view depending on the direction.

For example you could extended the field to include the corners when you travel in cardinal directions and shrink it down two squares on each end in your diagonal cases so that each uncovers 9 squares.

Another alternative depending on how your lighting works would be to use a better circle approximation with antialiasing, to partially reveal some of the squares.