Shifting hexagonal tile render location?

The above picture is what I currently have, where the tops of the tiles go beyond the grid.

The second picture is an artist’s impression of what I want it to look like, where the bottom extends beyond the grid.

Layer offsets doesn’t do this, I assume if there is a way to do this, it’d be in the tilemap, but I can’t find anything that would achieve this

In the Tileset’s properties, you can change the Drawing Offset, which controls how the tiles re rendered relative to the tile grid.

To access the Tileset properties, click the “Edit Tileset” button under the tileset in the Tilesets view, that’ll open the Tileset for editing or focus it if it isn’t in focus already. If the Properties view doesn’t show the Tileset properties already, you can open those via the top menus: Tileset → Tileset Properties…