Civic sacrifice did not answer every need, and a range of other practices grew up alongside it.
Oracles were consulted by states and individuals on questions of action rather than of doctrine: whether to found a colony, go to war, marry, or which god to placate. Delphi was the most prestigious, where enquirers received responses through the Pythia. Dodona in Epirus answered through the oracle of Zeus, and hundreds of lead strips carrying private questions have been recovered there, asking about children, theft, livestock and travel. The famous ambiguity of oracular replies is largely a feature of the literary tradition; the surviving questions are mundane.
The mystery cults offered something the civic calendar did not: an experience undergone individually, and a change in the initiate's prospects. The Eleusinian Mysteries of Demeter and Persephone were celebrated near Athens for over a thousand years and were open to almost anyone who spoke Greek and had not incurred blood guilt. The secrecy held: surviving accounts describe the procession and the preparation, not what took place in the hall. What initiates were promised is usually summarised as a better lot after death, and beyond that the evidence thins quickly.
The dead and the heroes received cult in their own right. Families tended graves and made offerings at fixed points in the year, and cities honoured heroes at tombs, real or claimed. A hero might be a figure from epic, a city founder, or a general killed within living memory.
Below all of this ran everyday practice that the literary sources mention only in passing: curse tablets folded into wells and graves, purification after birth, death or bloodshed, amulets, and freelance specialists selling all of it. Plato complains about such people, which is good evidence that they existed and did well.