There are two key areas I can think of where the military will have had a big imact on society as a whole in archaic Greece: tyranny and democracy. These are two forms of government which get some attention for this period as they led to massive changes in the way things were done, and who held the power, in archaic and early classical Greece.

1. Tyranny. In order to usurp power from the existing, usually aristocratic government of a city, a tyrant needed a means to seize and hold on to power. This was usually through military might, though popularity amongst the lower (non-aristocratic) classes and piety didn't hurt either. It was, for example, with an army that Pisistratus, the tyrant of Athens, made his third, successful attempt at seizing rulership of Athens. And it was while he was polemarch (head military guy) of Corinth that Cypselus expelled the Bacchiadae and became tyrant of Corinth.

2. Democracy. This goes over into the classical period, but basically, as Athens was developing its new (post-Pisistratid dynasty) democratic system, there were a number of class dictinctions. The important ones here are between the thetes - the poorest - and the hoplites - a sort of middling class. See, there was something of a power struggle over who could vote, who could hold certain offices, even who could be a citizen of Athens. After the battle of Marathon in 490 BCE, which was won by the phalanx of hoplite soldiers, it was that class of people who held the balance of power compared to the more numerous but less wealthy thetes, but after the battles of Artemesia and Salamis a decade later, which were both naval battles, the glory was claimed by the rowers - thetes. Class, or rather wealth, determined where someone fought because it determined whether they could afford the armour, shield and weapons of a hoplite soldier (or, indeed, the horse of the wealthier cavalry level) or not - thus meaning in war you'd be a rower in Athens' extensive navy.

To find out more about this, there's a section in Greece in the Making by Robin Osborne you can have a look at.