If you want to use the axial coordinate system, just set the grid to isometric, and paint hexagons on it.
It is what I did with my game.
The only problem currently is that Tiled automap and terrain features don’t consider hexagons to be adjacent when they should:
Actually, I think considering only the horizontal(if you hex are point up/down) OR the vertical (if your hex are point left/right) hexes adjacent would be the only missing feature to make the grid work “natively” for hexagonal hex (and replacing the tile shape with an hexagon).
That is mostly a problem when using automaps for me (as I don’t use the terrain tool at all).