So the three shows that I know who do this consistently are Super Sentai/Power Rangers and Kamen Rider (Super-Sentai's sister show. Concept is to Super Sentai/Power Rangers, but normally with a smaller cast of core heroes, and using a various unlockable forms rather than a large team of multicolored heroes).
The Japanese productions will typically run for one year with about 45-50 episodes per season. As such, the "Sixth Ranger" or "Second Rider" will typically be added into the cast sometime between the 10th and 25th episode (though early and later additions have been known to happen). In Kamen Rider's case, the new rider addition can be earlier because of the time of year the season starts (August-September) the new hero will be introduced at a time where the toys will be on the shelves for Christmas (Super Sentai typically starts seasons in Mid-Feburary so by Christmas, they are gearing up for the finale. However, because the Super Sentai season that introduced the Sixth Ranger would be the source material for the first season of Power Rangers, and the "Green Ranger" was massively popular in the U.S. market, they show will typically keep the Sixth ranger introduction early in it's season, since U.S. shows start in September-October, preserving the Christmas toy sale boost.).
From a story perspective, the somewhat early introduction is done as the new character creates a lot of story drama around the introduction and how the character shakes up the group dynamic of the core cast of characters. In Kamen Rider, the "Second Rider" is almost always an uneasy ally and while firmly against the villains, has different (and normally less noble or pettier) motivations from the hero of the season that will often cause disputes in tactics. In Super Sentai, the Sixth Ranger is not always guaranteed to be hostile to the heroes of the season, but will often serve to shake up the group dynamic of the heroes in some way, and is traditionally a foil to the team's red ranger (Usually the protagonist of the season). To compare the original Green Ranger and his Japanese counterpart, the U.S. Green Ranger was brainwashed into his evil and was able to strike the rangers in ways that the villains ordinarily could not, and frequently targeted the Red Ranger who was the team leader. In the Japanese story, the Green Ranger was never brain washed and his brief alliance with the villains was out of convience and he left them on his own accord. His grievances with the Red Ranger (and primary motivation) was due to being the Red Ranger's long lost brother and was bitter over the circumstances that lead to their separation.
In seasons where the Sixth Ranger is an ally from the start, he will typically work alone and once convinced to join the team will shake up the group dynamics as they may not understand the social cues of other members on the team. For a non-Power Ranger example, in Avatar: The Last Airbender, the first episode following Toph's addition to the group was an episode where the personality of her conflicted with that of the team (notably Katarra). Typically in this case, neither the older team members nor the new one are to blame, but rather the beginning team members are fire forged and all made their own mistakes together, so they are now working well and understand the way each of them interacts, while the new person has not been a team player and struggles with working with the established order... which is unworkable anyway because the established order only worked for a team that had less members than present now.
In Kamen Rider, there is also a different dynamic in that, a consistent theme is that the Kamen Rider(s) of a season draw their power from the same source as the villains, and thus are uniquely able to fight the villains that conventional forces lack. As such, most hero riders have some tragic element related to the villain's rise to power that currently makes them an active threat. The "Second Rider" will have a similar connection but might be more pessimistic and edgier than the hero, who has refused to let the tragedy prevent them from being an optimist and more understanding to the villain's plight or their victims. When it isn't that, the hero's civilian identity makes him a uniquely positive person compared to "second rider". For example, in Kamen Rider Ex-Aid, all heroic riders are doctors, but not the same specialty. The two most featured riders, "Ex-Aid" and "Brave" reflect their specialties in their attitude. Ex-Aide is a pediatrician, and thus much of his practice deals with calming down nervous patients and building a rapport with them. Brave is a surgeon, and because his specialty involves more risk to his patients, he actively avoids interacting with his patients because his attachment to them risks him getting emotional when they start flatlining under the knife and he needs to quickly fix the problem. When facing the villains, this is reflect with Ex-Aid being much more aware of the safety of those in the area, while Brave would be more likely to engage the monster and eliminate the threat. This results in conflict because Ex-Aid will work on solving the immediate problem of the victims at the risk of letting the bad guy get away, while Brave would rather stop the Monster quickly to prevent more people suffering like the current victims.
TL;DR: You should introduce the deuteragonist after you have your core cast dynamic established, but with enough story left so that you can establish a new cast dynamic with their arrival.
