You are referring to the Aeschylus fragment "Prometheus Unbound". Unfortunately this play was mostly lost.
(The source text can be found here: "The Prometheus bound of Aeschylus and the fragments of the Prometheus unbound" on page 145, but it won't be especially helpful unless you have some Latin and Greek.)
For the Prometheus Unbound fragments on Theoi, use this link and look for "PROMÊTHEUS LYOMENOS".
I'll amend if I find any further information, but it looks like any parts of this text that may have told what happens to Prometheus after he is unbound in this play are lost.
However, there is a third play, Prometheus the Fire Bringer, again, surviving only in fragments and only. On Theoi, use this link and look for "PROMÊTHEUS PYRPHOROS".
One theory on this third play hold that Prometheus is reconciled with Zeus after revealing to Zeus the prophecy of Thetis' child becoming greater than Zeus himself [see Brian Donovan's comment below for more clarification], which may derive from Hyginus, Fabulae, 54, and is predicted by Prometheus himself in the Aeschylus. (If true, this 3rd play might have contained the answer on what Prometheus does after he is unbound, but there is an alternate theory holding that this play was actually the first in the trilogy, and depicted the events concerning the theft of fire.)
Graves says that
"Mankind now began to wear rings in Prometheus' honour, and also wreathes, because when released, Prometheus was ordered to crown himself with a willow wreath."
Source: Robert graves, The Greek Myths, 133.m, citing Athenaeus
so there's that.
For more info on this punishment and Athenaeus, see "The Author of the Prometheus Bound" by C.J. Herington.
RELATED: Prometheus Unbound by Percy Bysshe Shelley
Note: Shelly is not part of the classical canon, but is a poet of sufficient stature to be included in the overall literary canon, so he is worth reading if you have an interest in this subject.