Excellent!

Quick clarification for further question-asking: "Hellenistic" typically applies to the period of Greek culture between the death of Alexander the Great and the beginnings of the Roman Empire. "Hellenic" refers to ancient Greece and its culture prior to ATG.

It's also important to remember that in Greek (as well as in Semitic languages), poetry predates prose. So the older material in these cultures is often poetry.

I'm less well versed in Hellenistic literature, but you could look into Apollinus of Rhodes. He's famous for writing the Argonautica. He straddled the line between Hellenic and Hellenistic.

In terms of Hellenic poetry, there are (in my opinion) many more interesting options. Everything written by Homer falls into the realm of Epic Poetry and is written in dactylic hexameter. [Side note, I worked through an entire book of the Odyssey in greek this past fall and it was awesome/miserable]. There are tons of Hellenic Greek playwrights to read as well. Again, this is just my opinion, but there is no going wrong with Aeschylus. The Persians is extremely engaging and all three portions of The Oresteia are absolutely wonderful. I'd also recommend Antigone by Euripides. Round it out with some Sophocles and you're set.

Interesting note, all three of them lived at the same time and their works were performed in competition with one another.

Hellenistic verse is characterized generally by a break from older forms of Greek verse. As opposed to the relatively free distribution of poetry and poets in the Classical period, the Hellenistic period saw writers clustering increasingly at specific points. This was to a certain extent true in the Classical period, in which Athens was the preeminent literary center, but sites like Alexandria and Antioch tended to attract writers because court patronage would provide the ability to steadily produce. Hellenistic literature in general, and verse in particular, became increasingly esoteric, as it was the product of highly-educate scholars whose literal jobs were to study and produce verse.

We can roughly divide the major Hellenistic verse forms into dramatic, satiric, and elegaic verse, based on thematic material, publication, and of course meter. In the Classical period dramatic verse as we understand it was pretty limited to Athens, and comedy in particular was a uniquely Athenian form. Already by the end of the Peloponnesian War Old Comedy had begun to transition into Middle Comedy, which was a short-lived form that quickly developed into New Comedy, which featured simpler plots (usually stock, and avoiding the political themes that were so usual in Aristophanes' Old Comedy). By the time of Menander, the great New Comedy poet, the comic chorus had all but disappeared entirely. The Hellenistic period saw New Comedy's spread throughout the Greek world, and though tragedy continued to be produced it was increasingly an old and stuffy form. The third century was extremely prolific in New Comedy, but unfortunately next to nothing survives. A single play, Menander's Δύσκολος, is known almost in its entirety thanks to a fortunate papyrus find, and we have significant fragments of some seven of his other plays. We have fragments of Diphilus, Philemon, and some other New Comedy, but Menander is our only lengthy survivor--and, of course, the greatest New Comedian. Fortunately for us, the Latin playwrights Terence and Plautus adapted quite a large number of New Comedies into Latin. Their adaptations are often quite free when compared with our fragments, but they give an idea of what the complete pieces were like (New Comedy was not particularly inventive in terms of plot and characters, relying on stock versions of both), and they're incredibly amusing and beautiful pieces of verse in their own right.

The Hellenistic period also saw the development of iambic verse, which we usually call satire because of its later Latin development in Horace and Juvenal. Iambic verse (that is, verse written in iambic meter) goes back at least to Archilochus, but its golden period is in the Hellenistic period. The defining feature of iambic verse, other than its meter, is that it's characteristically abusive. It attacks and satirizes people (including the poet), often quite mercilessly. Several Hellenistic writers of iambs are known, and a fair number of fragments have survived. Callimachus wrote a number of satires, and Timon, himself a philosopher of the skeptic school, poked fun at philosophers in his poems. Cercidas wrote iambic verse in lyric meter, and according to the Suda one of Sotades' iambs attacked the Ptolemies, the last Greek poet to include major political themes openly in verse.

Mention of Callimachus brings us to elegy and other non-epic hexameter verse, which is the verse form of the Hellenistic period. Callimachus was a scholar at Alexandria, and was, without a doubt, the greatest Hellenistic poet. Most of Callimachus' prodigious output is lost, but he was influential enough to bring us the neoteric poets in Latin, the circle of poets including Valerius Cato, Catullus, Licinius, Calvus, Cinna, who in turn influenced Horace, Virgil, Propertius, and Ovid. Callimachus did not invent elegy or the use of hexameter in hymns and other forms besides the traditional Homeric epic, but under his leadership it became the great form of Hellenistic literature. Fragments of his Αἰτία ("Origins"), Ἴαμβοι (iambs), Hecale, Hymns, and a good sixty-something epigrams survive. Callimachean verse was rather eclectic, and he wrote on a great number of themes. By and large his stuff was hexameter, often elegaic--notably, he did not write love elegy, which was a Latin development. The Hecale is indicative also of a new, Hellenistic, type of hexameter verse, the epyllium, a sort of shortened epic (of which the word is the diminutive). Epyllia predated the Hellenistic poets technically, but only really took off under Callimachus and his successors, who tried to jam an entire epic into a few hundred lines. Often this forced the poet to introduce staccato lines or refer to things obliquely--short hexameter forms like this provided the poet with ample opportunity to show off his grasp of little-known mythological subjects or historical events, as well as letting him show how good he was at the form by packing as much meaning as he could into as few lines as possible. Only fragments of the Hecale survive, but Catullus 63 is a Callimachean epyllium about the wedding of Peleus and Thetis, and like much elegaic verse it quickly diverges from its supposed theme and starts talking about Theseus' abandoning of Ariadne instead. Other hexameter forms included the didactic epics of Aratus and others, which are essentially just prose works rewritten for epic verse, and Theocritus' bucolic verse, which provided the model for Virgil's Eclogues, although like a lot of Latin verse they rapidly diverge from their models. "Proper" epic did not die out in the Hellenistic period--Apollonius Rhodius' Argonautica is at least Homeric in form, if not necessarily in content--but it generally diverged from the heroic model of Homer and ventured more into purely mythological or historical subjects. Little Hellenistic epic survives, and poets of the period generally preferred the shorter forms, which blossomed under the Alexandrian scholars.

That doesn't really do justice to Hellenistic verse, which, though you might (I certainly do) quibble with its actual quality was extremely influential and inventive. It would take a book to adequately lay out Hellenistic verse in full, and reddit doesn't have the ability. As good a treatment of any is Hutchinson's Hellenistic Poetry, which is basically just a textbook overview. Hellenistic poets, and particularly Callimachus, were extremely influential on Latin poets, however, and although relatively little Hellenistic verse survives we can often understand it through our much larger body of Latin neoteric and Augustan verse.