Not by using the grid command, no. The "even columns are higher" option is just for hex grids due to the way hexes work, they have to alternate, but it is not a generic feature for offsetting grids.

It is too late for me to go into detailed explanations right now, but you should be able to do this by setting up a symbol fill. Symbol fills use repeating patterns, so you just need a fill style symbol (which is a regular symbol) that captures the minimum repeating pattern you will need, probably a 2x2 grid square pattern, but with the left column offset.

Give me a word if you can't figure this out by yourself and no one else replies with the instructions and I'll come back with more details when I have time and is awake.

Thanks @Wyvern, these are now in the atlas.

These maps are now in the atlas. Thanks @Royal Scribe

The map has now been integrated into the atlas. Thanks for a great submission @HelenAA

Nice. That last one is great.