The series follows Captain Christopher Pike (Anson Mount) and the crew of the Enterprise years before the events of Star Trek: The Original Series. As Pike's fate is already known, the commission will eventually go to James T. Kirk (Paul Wesley). The crew of TOS is being introduced a piece at a time, slowly inching towards the legacy show.
In an interview with ScreenRanti's Joe Deckelmeier, showrunners Henry Alonso Myers and Akiva Goldsman were asked about the future of the crew of the Enterprise after Strange New Worlds ends, and if some TOS stories would be retold with the new cast. Myers was coy, not confirming anything. Goldsman notes that they are interested "in telling stories of Kirk's Enterprise that we might not yet know about."
ScreenRant: At Strange New Worlds season 3's premiere, you spoke about continuing the show into The Original Series era with Kirk and Spock. Is that feasible? When Strange New Worlds ends, are you going to go directly into TOS and retell stories with that original crew?
Henry Alonso Myers: We can neither confirm nor deny.
Akiva Goldsman: We are very interested in continuing to tell stories... I'm gonna keep trying to find a good quote here. We are very interested in telling stories of Kirk's Enterprise that we might not yet know about yet.
What This Means For The Future Of Star Trek
The TOS Crew Is Slowly Being Introduced
Strange New Worlds debuted with a few TOS characters already on the Enterprise, including Lieutenant Spock (Ethan Peck), Cadet Uhura (Celia Rose Gooding), Nurse Chapel (Jess Bush), and Dr. M'Benga (Babs Olusanmokun), whose character was in two episodes of TOS. Since then, Lieutenant Kirk, and Lieutenant Junior Grade Scott (Martin Quinn) have been added.
If the new cast were to continue into a new series, they would have room to do so before the events of TOS' first episode.
A few TOS characters still need to be introduced, including Dr. McCoy, Hikaru Sulu, and Pavel Chekov. While Chekov was not introduced until TOS season 2, he refers to being present at events from season 1 in Star Trek II: The Wrath of Khan.
When TOS began in 1966, the Enterprise was on its five-year mission. The series drops viewers into the middle of the action in season 1, episode 1, "The Man Trap." This is clearly not the first mission under Captain Kirk, so if the new cast were to continue into a new series, they would have room to do so before the events of TOS' first episode.