Ancient Greek tragedies tended to be written as trilogies with a satyr play. We unfortunately only have one trilogy -- Aeschylus' The Oresteia, which includes Agamemnon, The Libation Bearers, and The Eumenides. However, we know that this structure was common, even if we do not have the complete set for any of the rest of the plays we have.
Thus, we can assume that Euripides' play Hippolytus was one of a trilogy. In fact, we can safely assume it was the first of the series. We can do this because it seems clear that there is nothing that came before which we need to understand, while the end of the play provides a clear indication that Artemis is going to do something to Aphrodite in revenge for what Aphrodite did to Hippolytus.
If we consider the one intact trilogy we have, we can work out what was the likely outcome of the tragedy (if not the final satyr play).
Hippolytus begins with Aphrodite saying she is going to avenge herself on Hippolytus because he disrespects her and curses her. The action of the play is thus caused by Aphrodite's desire for revenge. To accomplish her revenge, she causes Phaedra to fall in love with Hippolytus. Phaedra feels guilty for being in love with Hippolytus, but this guilt is turned into shame when her Nurse, in whom she confides, decides to tell Hippolytus, believing it is better Hippolytus knows than that he doesn't. Of course when Hippolytus learns of Phaedra's feelings, he curses the Nurse for telling him. But the Nurse had wisely asked Hippolytus to keep silent regarding what she was going to tell him, so he will not tell anyone. However, Phaedra is under the impression that he is going to tell her husband, his father, Theseus. The guilt she feels, combined with the shame she feels regarding Hippolytus' knowledge of her feelings, combined with the shame of his rejection, combined with her anger at him for cursing all women leads her to want to avenge herself on Hippolytus. More, she does not want her shame to spread to her children, so she has to silence Hippolytus. She thus writes a letter claiming Hippolytus raped her, and commits suicide to punctuate the letter, thus ensuring Hippolytus won't be believed no matter what he says. Thus will her reputation remain intact.
Committing suicide to protect one's reputation may, for someone who feels guilt rather than shame, seem an odd thing to do, but given the connection between shame and reputation, and the fact that one's shame can spread to one's relatives (while one's guilt cannot), her suicide does in fact make sense. It would have made sense to the Greek audience who first saw the play.
When Theseus discovers the letter, he curses Hippolytus in revenge. Once that revenge is realized, a dying Hippolytus forgives Theseus and Phaedra while Artemis tells Thessus what happened and that she will now avenge herself on Aphrodite's favorite.
Given that Adonis' fate -- to spend a third of his time with Persephone and a third of his time with Aphrodite and a third of his time being his choice (he chose Aphrodite) -- had previously been decided by Zeus, we would not be surprised if the third play involved Zeus setting up a permanent celestial court to decide conflicts between the gods. After all, by Artemis killing Adonis, Zeus' decision regarding Adonis is nullified -- Persephone now gets Adonis permanently. One cannot imagine that Zeus would have been favorable to that outcome, meaning he would have an incentive to ensure that his will were not again circumvented by one of the other gods. Further, given that Aphrodite goes into Hades after Adonis, Zeus would have to again decide what to do with Adonis -- which turns out to be to split his time with Aphrodite and Persephone equally. Of course, Artemis, too, sought the resurrection of Hippolytus, and she would have likely insisted that if Aphrodite could get Adonis back, she should be able to get Hippolytus back. Hippolytus is, of course, resurrected, and Artemis makes him her priest.
Thus, a celestial court emerges that ends the revenge cycle. This would reflect those tragedies that celebrate the establishment of the earthly court system of Athens, such as we see in The Oresteia.
It is unfortunate we have no idea what the satyr play could be in either case.