The algorithm should work just as well for player locations that are "in between" tile centers. For instance, a player at (1.25, 4) would be 25% of the way from (1, 4) to (2, 4). Their field of view could be computed as if they were at (1, 4), except that the shadow angles would be different.

This could be used for smooth transitions as the player moves from one tile to another.