In a series, I expect to find one of three approaches:
a) One single plot that covers the entire series and which is divided into smaller parts in order to give each book some level of closure
b) One plot that covers the entire series, and smaller plots that start and end in each book.
c) small plots that start and end in each book, with the unifying line of the series being a location or something else.
So the first step is to identify which approach you're following.
The plot is about the MC's coming of age (becoming an adult). But one does not become an adult as a result of a single experience. Perhaps the MC started out as insecure and found confidence, but there is still a lot to learn in order to become a full fledged adult. The next book will simply be another step in that long travel.
In this case, there is a major plot that requires the MC to come of age in order to successfully tackle it. So it makes sense for each book to have a slightly different tone, though not necessarily. The attitudes of the MC can change, as her worldview, but the tone doesn't have to become grim.
From your words...
the plot does not end with the current novel. A major arc is resolved, but many themes are left open for a potential sequel.
...I assume 'approach C' is your situation.
If the readers are after 'coming of age' themes, then there's only one possibility: turn this into option A and simply say it's the next step in her growth. This way, each book tackles one difficulty in the process of becoming an adult: responsibility for one's own actions, developing skills essential to an independent life, romance and sex, financial difficulties, dealing with a job/boss... anything that fits the setting and separates a dependent, inexperencied youth from an independent, experienced adult.
But the series can be presented as being something else. This could be the adventures of a woman, with the first book explaining how she became that independent woman. In that case, the reader knows that the first book is simply the setting to the real story in the upcoming titles.
In my understanding, people purchasing book 2 of a series do that because they're happy to read the follow up to book 1.
I completely agree with you.
Yet, leaving behind the "coming of age" theme can be considered as breaking a premise.
So do not present 'coming of age' as the premise of the series. It's simply the first theme of the series, with other themes showing up in later titles. If you do not trick your readers by letting them think this is all about growing up, they'll carry on with you.
What happens after the classical coming of age story?
In Historical Fiction, it's not uncommon for the main character to start out as naïve and then grow up into a powerful - often cunning - person. The difference is that the focus isn't in the coming of age. The focus is in finding out how the character got to the top, and then how they faced their enemies in order to remain at the top... and whether they were successful or not.
Most stories marketed as 'coming of age' tend to avoid this wide focus. The character faces the challenges successfully and becomes a grown man/woman. The End.
Can you rule out the "growing character" aspect without losing and alienating a part of your audience?
Most readers (IMO) who gravitate to these tales are after that victory and don't really care about the later adventures. They want to experience those growing pains and bask in the final victory. If that is the target reader (typically a teen or young adult), then there really aren't many alternatives: either stretch the 'coming of age' and explore all the themes you can find within it (which are plenty, to be honest) or don't have a sequel.
But if you are after a different audience, an audience who is fed up with 'coming of age' and who scoff at the 'happily ever after' finale, then pitch the 'coming of age' as simply the first adventure and give plenty of warning that things will carry on with other themes. Or avoid starting out with the coming of age. Let the readers get to know and care for the skilled adult character before they have to put up with a clueless kid learning the ropes.
PS: I almost didn't write this, but then decided to give my personal take on the topic.
I'm terribly fed up with the clichéd 'coming of age'. I like the idea - I love it, really! - but it never goes beyond that first victory and the 'happily ever after'. A character that becomes adult and keeps on having adventures sounds far more interesting, though the narrator mustn't strike too much of a different tone (IMHO).