disabling mip-mapping
Disabling mip-mapping is not the best idea. For one you will get strong artifacts when moving around and using detailed textures, and it is slower than using mipmapping.
Whilst this seems to work (through trial and error) is there any logic to this?
Yes, the reason is, that on a atlas the naive pixel coordinates are often between two adjacent textures. Therefore the samples taken at the border will include texture data from the adjacent texture (reason: linear filtering). To reduce the effect, you need to move your coord away to the center of the border pixel which you have done by moving it by a whole pixel away (half would be enough).
This could be done with lower mipmapping levels too, but in this case you need to move it away depending on the mipmap level, that is
level 0: 1/width
level 1: 1/ (width/2) = 2/width
level 2: 1/ (width/4) = 4/width
level 3: 1/ (width/8) = 8/width
...
As you can see, higher mipmap levels result in a relative broad border, so you can't just calculate it once and apply it to all levels. Either you need to ajdust it dynamically depending on the mipmap level (expensive, newer hardware needed) or you should repeat the border of your tile. As far as I remember the latter has been done in the id-tech5 engine (megatextures utilize lot of atlases).