Gap in the tiles when using 3x2

Not if you are using separate Tilesets.

Each Tileset force starts a new row.

There's no way that Tiles from the 3rd Tileset can intermingle with the 2nd.

I am trying to achieve something like this:

4cf016e036ef870d5ecaed704308684cad4eaa32.jpg

How do I do that?

It comes down to trial and error...but maybe start by, moving the "mode tile" after the weather tile and increase the size of your tiles on the panel (in the panel style settings).