Tiled level layouts do not mean movement has to be from tile to tile. You can still have positions stored as float, and adjust position gradually within a tile, without having to snap to up/left/right/down tiles on movement.

In a topdown (non-isometric) level, with each tile being e.g. 128x156 pixels in size, you can move around as if the level weren't tiled. You can, however, use the tile system to figure out e.g. collisions.

Using some simple math you can figure out which tile you're standing on, and only check for collisions or whatever in tiles that are adjacent (or otherwise interesting).

In this case, you could figure out the tile by using pos.x / 128 and pos.y / 156.

This also works for isometric layouts, but the math is (slightly) more involved.