The biggest reason NOT to go north-to-south is the rain cycle.

The rainiest two months of the year in Montana are May & June.
Plus, at high elevation there is still plenty of snow and mud.
By late July it is significantly drier in the Northern Rockies.

Conversely, the monsoon season in New Mexico is July & August.
And the clay/caliche soils of dirt roads turn into super-glue.
May and June are usually quite dry - the end of the dry season.