As an alternative to a more complex field of view (which as discussed above adds its own problems because of the constraints of a grid-based layout) you could try to emulate the effect of movement in a game that isn't based on a discreet grid. Where free movement is possible a diagonal move of one unit would be exactly that, not the ~1.41 units of movement seen with a square grid.
While you can't force single unit movement without losing your descreet grid (which would change your game design pretty significantly) perhaps you could keep track of the extra movement taken and drop moves later: track the extra 0.41s and once they total more then 1.00 have that unit skip a move. Or the other way around: consider diagonal to be normal, add up the 0.41s for each horizontal or vertical move, and give extra move credit once that is greater than 1 (or 1.41 for a diagonal move).
You would need to be careful how you present this to your players in a way that makes it look both smooth and fair. In a multi-human-player scenario such changes could become something the players take advantage of strategically - this may be a problem, or there may be a natural way to mix this in with the game mechanic (perhaps allow the players to store a small amount of "unused move credit" that they can use to react quickly at a later time and have the extra 0.41s of move feed into (or take out from) that pool.
This would work best if player control entities moved move than one unit each turn. For example three movement points might be used as three horzontal moves, or two diagonal ones with 0.16 left in the pool for later. Once that hits 1.00 the player gets a "free" hor/very move and at 1.41 a free diagonal one. You could cap the extra at 1.5 for force it to be used or lost at that point to stop the player keeping this stored energy for ages or letting it build up.
Obviously this is a complication to your game rules that may be completely undesirable, and it would be impractical for a non-computerised game, but if you could make it work within your game's existing rules it would limit the exploration difference between movement directions without needing to abandon the grid format.