To say that the snake stands for wisdom is to gloss over many of the meaning attributed to the serpent in the world's mythopoeia — the serpent is dual-aspected, and can stand either for good, or for evil, depending on the cultural context.
Context, as ever, is everything. To assume a 'one-definition-fits-all' is mistaken.
There is the serpent/tree relation in Scripture, in Greek mythology, in Norse mythology, in Mayan mythology ...
The Buddha sat beneath a tree, and when a storm arose, the serpent Mucalinda rose up from his place beneath the earth and enveloped the Buddha in seven coils for seven days, not to break his ecstatic state.
What is relevant however is that it is not the serpent who imparts knowledge — the serpent protects the tree at which knowledge can be found.
Mentioned above is the psychoactive element (derived from the effects of snakebite) such as the 'Vision Serpent' of the Mayans. But the use of entheogens belong to the Rites of Eros, where the self is overthrown or carried away in some other way possessed — the story of Eden happens 'in the cold light of day'.
Common to many systems, 'the tree in the midst' or the 'world tree' indicates the vertical axis, whereas the serpent then signifies movement on the horizontal plane, a serpent being the nearest thing to a 2D entity.
Thus there is a link in many cultures where the tree/serpent configuration represents communication between the spiritual and the earthly worlds or planes.
Often there are two serpents. The Sumerian deity Ningizzida, is accompanied by two gryphons Mushussu, the oldest known image of two snakes coiling around an axial rod, dating from before 2000 BCE. The coiled snakes of the caduceus of Hermes, for example, or the Rod of Asclepius, the staff of Moses ...
An interesting correlate is the Flood — Ningizzida was the ancestor of Gilgamesh, who according to the epic dived to the bottom of the waters to retrieve the plant of life. But while he rested, a serpent came and ate the plant. The snake became immortal, and Gilgamesh was destined to die.
The Hindu concept of Kundalini (Sanskrit: "coiled up" or "coiling like a snake") refers to the intellect and spiritual maturation. Joseph Campbell has suggested that the symbol of snakes coiled around a staff is an ancient representation of Kundalini physiology — the staff represents the spinal column with the snake(s) being energy channels. In the case of two coiled snakes they usually cross each other seven times, a possible reference to the seven energy centers called chakras.
In a more 'esoteric' reading from the Hermetic Tradition, the two serpents represent the dual-aspected movement around the vertical axis or Principle. This goes on to the imagery of the eagle clutching the serpent ...
In Ancient Egypt the serpent appears from the beginning to the end of their mythology. Nehebkau was the two headed serpent deity who guarded the entrance to the underworld. The primal snake goddess Wadjet was the patron and protector of the country, all other deities, and the pharaohs. Hers is the first known oracle.
The Basilisk, the "king of serpents" with the glance that kills, was hatched by a serpent.
Typhon was the enemy of the Olympian gods is described as a vast grisly monster with a hundred heads and a hundred serpents issuing from his thighs, who was conquered and cast into Tartarus by Zeus, or confined beneath volcanic regions, where he is the cause of eruptions.
Python was the earth-dragon of Delphi, she always was represented in the vase-paintings and by sculptors as a serpent. The enemy of Apollo, who slew her and remade her former home his own oracle, the most famous in Classical Greece.
Not to mention the Gorgons, who wore a belt of two intertwined serpents in the same configuration of the caduceus. Gorgon blood had magical properties: if taken from the left side of the Gorgon, it was a fatal poison; from the right side, the blood was capable of bringing the dead back to life.
And Ouroborus ... which signifies constant cyclic movement on a single plane, although that plane can be, for Plato, a kind of meta-plane containing all planes ...
God bless
Thomas