Actually Sparta's army had also slingers, a role reserved to helots.
Would be also very cool to consider a new unit based on the figure of the "aulos" players. Aulos is a sort of flute, and those players had the role to guide and keep orderly the formations of hoplites through their music. They were highly respected among the population.
They should have an aura and could give some bonuses only to infantry set on formation. Something similar to the Trumpeters of the Gauls. "It was a sight equally grand and terrifying when they marched in step with the rhythm of the flute, without any gap in their line of battle, and with no confusion in their souls, but calmly and cheerfully moving with the strains of their hymn into the deadly fight. Neither fear nor excessive fury is likely to possess men so disposed, but rather a firm purpose full of hope and courage, believing as they do that Heaven is their ally."
Plutarch, The Parallel Lives (Chapter II, section III)
"And then there was the clash: the Argives and their allies advanced impetuously and full of anger, the Lacedaemonians instead advanced slow and at the cadence marked by numerous flutists placed among them not for reasons of religious worship but to be able to march in step with that rhythm and so the ranks would not get confused, which often happens to the great ones when they go to the assault "
Aulio Gellio, Noctic Atticae (I, XI, 1-5)
Could also be considered to add a new figure to the Temple: the oracle. Among ancient greek civilization was very common to make animal sacrifices towards the gods before engaging a battle, to determine their will.
We could add the possibility to train this figure inside the temple, who can use hunt to make those sacrifices. These can give positive or negative response, in a completely random way, with a consequent temporary bonus (or maybe a malus if response is negative) for the army.