here is an old post of mine about Travelling Post which may help you understand that system, Line.

If you hired horses you might not hire a coachman. Normally, he would be part of your permanant staff if you owned a carraige, along with postillions.

See this about the duties of the postillion travelling when "post" from The Compelte Servant by Samuel and Sarah Adams(1825):

The Postillion.

When the family travels post, the helper in the stables, and the stable boy, generally ride as postillions: on other occasions, the boy or helper, rides and drives the chariot and other carraiges; or if he be a regular servant, he has the care of a pair of horses
Page 164.

You might like to see this rather wonderful drawing by Thomas Rowlandon of the process of hiring a post chaise from an inn ;-)