Having given this some more thought, however, I realise that my list is actually missing characters from what I think is an often misunderstood component of the war, namely the Gigantes [Giants]. Many mythologists are puzzled by what they often interpret to be a confusion of the two greatest wars in which the gods fought, i.e. the Titanomachy in primeval times and the Gigantomachy in the time of Herakles (a generation before the Trojan War). And for certain at least one ancient writer, the Roman Gaius Hyginus, does confuse the two a whole lot, just like he mixes up a lot of the characters in his Greek mythology. There are, however, other ancient authors who make plenty of explicit references specifically to Gigantes fighting against the gods during the Titanomachy, and many modern scholars, confused by this, would conclude that whichever mythographer is reporting these battles is confusing two different wars. I, on the other hand, am going to use that to update my response to the original question, since it now seems clear to me that most of those ancient mythographers, rather than being confused, are perhaps making reference to the now-lost portions of the story of the conflict between Kronos and Zeus which would have been detailed in lost epics such as the Titanomakhia [Titanomachy] by Eumelos of Korinthos or Arktinos of Miletos.
Throughout the course of Greek mythology’s timeline, there are several births of Gigantes, instances of a variety of different groups of them springing up from the Earth. There was at least one group which was born before Zeus’ time, a number of groups who came about in between the times of the Titanomachy and of Herakles, and then there’s the famous crop which was destroyed by the gods during its major attack on Olympus later called the Gigantomachy. Before Zeus was born, there was a squad of Giants who wore full suits of armour, carried an arsenal of weapons and were led by a certain Hoplodamos. Rhea recruited these guys as her bodyguards while she was pregnant with Zeus, to protect her from her own husband Kronos. When the time came for Rhea to deliver her baby, Kronos showed up ready to devour the newborn as he had done with the previous five, but Hoplodamos and his brothers attempted to deny him access. A violent battle broke out between Rhea’s husband and the Giants, and Kronos slew all the bodyguards. Nonetheless they had given Rhea enough time to swap the baby for a stone which she managed to convince Kronos that it was his child.
By the time Zeus had grown up Kronos himself had taken to conscripting Gigantes, this time for all-out war, and one of which Giants was called Sykeus. Other Gigantes whom Kronos may have drafted for the Titanomachy include Azeus, Damysos and Mylinos, the last of whom Diodoros Sikoulos seems to describe leading a rebellion against Zeus on Crete Island, where he and “his followers,” as Diodoros calls them, were destroyed. Perhaps some of the battles of the great war took place on Crete. Or could Mylinos and other Gigantes have split off from Kronos’ army to form their own faction off the mainland, opposing both the Titan and his son? The case of the rebel Olympos which Misa has mentioned is an interesting one. I wonder if Olympos raised Zeus when he was a baby and then betrayed him later on to join Kronos. Or perhaps once Zeus had become king and Olympos failed to see a difference between his tyranny and that of Kronos, he revolted against his own foster-son(?).
These are updates to my list, with individuals who I’d neglected to consider before:
13. “Ekhidna’s son” – An unnamed “dragon-footed” Gigantos whom Nonnos, in the Dionysiaka, describes as the enemy of the gods, spitting venom into the Sky in such profusion that it created large clouds of poison, and who was so huge that the birds flying through the air would get caught in the vast mass of his dreadlocked hair, which hair he would then use to sweep these birds into his mouth to chew them live for dinner. Nonnos seems to be describing a peculiar apparatus in which Kronos used this Gigantos as some sort of missile or shield to counter Zeus’ thunderbolts. The Gigantos was killed by Ares.
14. Sykeus – Athenaios’ Deipnosophistai [“Philosophers at Dinner”], quoting Tryphon’s History of Plants, quoting The Farmer’s Handbook by Androtion, specifically refers to Sykeus as “one of the Titans” whose mother was Ge [Gaia], essentially suggesting the idea that there was a 13th Titan who was the child of Ouranos and Gaia, although Ouranos is not explicitly mentioned in this passage. I’m thinking that rather it is a conflation of terms and that while Sykeus was indeed Gaia’s son, he was not a Titan but rather a Gigantos who fought on the side of Kronos' Titans. (I’m borrowing part of this idea from Aaron Atsma, the author of the Theoi Project website, who seems to interpret this passage to mean that Sykeus is one of the Gigantes from the Gigantomachy, which would only mean, however, that the authors quoted by Athenaios are among those who conflate the two great wars.) Sykeus was pursued by Zeus and hidden by his mother Gaia [within her own depths?], who caused the fig-tree, sykea, to grow “for her son’s pleasure”. The city of Sykea in Cilicia was also named after him.
15. Azeus(?) – A fragment from an unnamed writer says that a Gigas [or Gigantos] named Azeus son of Gaia “grew to manhood amid the mighty battles of the Titans.” The most natural reading of this is perhaps that Azeus just happened to turn 18 or 21 while the Titanomachy was raging and not necessarily that he participated in it at all. Then again perhaps the mention of battles means that he did actually take sides, joining either Kronos or Zeus. Perhaps he was one of the defectors described by Diodoros Sikoulos…
16. Mylinos(?) – A Gigantos who Diodoros Sikoulos describes as a leader among the “robbers and impious men” killed by Zeus on Crete Island during the Titanomachy. Presumably this means he sided with Kronos, unless he had his own island operation on the side (pulling heists?), concurrent with the war.
17. Damysos(?) – the fastest runner among the Gigantes
18. Olympos(?) – A Cretan who may have been a Gigantos or a Kourete [Curete] (or a Giant Kourete?), who raised Zeus when the god was still a little kid, teaching him “divine things”, whatever those were. For some unspecified reason Olympos incited certain Gigantes to attack Zeus and was slain for it. There is no direct reference in this story to more famous characters from the Gigantomachy, so perhaps Olympos’ attack should rather be associated with the Titanomachy, similarly to Mylinos’ operation?
19. Kampe – A cosmic-sized female monster of the Underworld, who was composed of tens of different animals, whom Nonnos’ Dionysiaka describes as a massive millipede-like creature. Technically she was not a soldier; before Zeus was born Kronos had made her the jailer of Tartaros, to guard the entrance to the cosmic pit where he had imprisoned his younger brothers the 3 Cyclopes and the 3 Hekatonkheires. Towards the end of the Titanomachy Gaia told Zeus that he would need the help of these six uncles of his if he was to defeat his father, so he descended into the Underworld, fought against Kampe, killed her and released the six prisoners.
20. Aigaios* – Surviving fragments of the Titanomakhia seem to say that one of the 3 Hekatonkheires, namely Aigaion, who is also called Briareus, was actually some sort of Sea-Titan (or Sea-Giant) named Aigaios, who was a son of Pontos [the male personification of the Mediterranean & Black Seas] and Gaia, or of Thalassa [a female personification of the Mediterranean]. In the epic, Aigaios/Aigaion, after whom the Aegean Sea is named, is an ally, not of Zeus, but of “the Titans”, meaning Kronos most likely, although, since we know there were defections, perhaps after being released from Tartaros, the Hekatonkheiros Aigaion-Briareus actually joined Kronos’ camp before changing his mind and supporting Zeus after all. Briareus does in fact become the most important Hekatonkheiros in the surviving mythology, being wed to Poseidon’s daughter Kymopoleia, apparently as a reward for his support of Poseidon and Zeus and their siblings in the Titanomachy.
21. Akheron - A river which used to flow in the upper world before it became an Underworld river (so neither a Gigantos nor a Titan, although Akheron was one of the 3000 sons of the eldest Titan Okeanos). Akheron was not a soldier, he simply supplied Kronos' team with drinking-water during the war. After Kronos' defeat, Zeus muddied Akheron's waters and cast him into the Underworld. Akheron's now-empty riverbed was eventually refilled but exclusively with the tears of dead souls suffering punishment for their sins in the pit of Tartaros, for which reason this river was so salty. Akheron, now an infernal god, became one of the five main rivers of the world of the dead.
22.—? Other Gigantes whose names we don’t know – Kronos may have had tens or maybe even hundreds of these at his disposal. They seem to generally have been mortal… and expendable, and thus generally to not have survived the war to receive the same punishments as Kronos and the other Titanic members of the losing team.