Correspondence - Study after apprenticeships
'Linked Fields' and Moving Between Disciplines
I think of the progression from apprentice to journeyman to master a little bit like the movement from undergrad to postgrad to someone with their PhD. (Obviously, apprenticeships are a real thing that did and do still exist, but I spend more time in academic contexts, so the comparison is helpful.)
In Harry's case, he has completed his apprenticeship, and can now call himself a Journeyman Snake Handler. Snake handling is a subset of beast handling, and it's not uncommon for people to do an apprenticeship in one area of beast handling, and then broaden or specialise or move sideways.
"The thing about Beast-Handling, see, is that there’s such a variety of animals that you’ve basically got to start with a general specialty and branch out from there. The Menagerie runs apprenticeships in large animal handling, with enough broad experience that you can start as a journeyman straight away in a couple of other fields. Did you know you can’t apprentice in dragon-taming? You can only study them as a journeyman, because they’re so dangerous. So if I finished an apprenticeship here, I could do a journeyman year on dragons, or I could go somewhere else and study freshwater creatures, or marine, or small animals, or ones from a different part of the world. Or I could do animal healing or something, but I think I prefer the heavy work. " - Whatever Happened to Draco Malfoy?, Chapter 18.
From where he is, Harry could try to broaden out into general beast-handling, or move sideways to work with another species group (like owls), or even go on and study dragons. Having completed a beast-handling apprenticeship, he can be trusted to have a basic understanding of how animals work and how you work with them, even though his species-specific knowledge won't be equivalent to someone who did an apprenticeship in the relevant specialisation. As such, he'd be a bit more closely supervised and require a bit more of a training period than journeymen generally, but he wouldn't be expected to go through a whole second apprenticeship.
If, on the other hand, he decided to study Charms at a higher level, he would need a second apprenticeship, and would list his qualifications in that separate to his qualifications in beast-handling.
Journeyman to Mastery
A journeyman, at the heart of the word, is simply someone who is trusted to be competent enough to hire on as a worker. They can work independently and reliably. Mastery, by contrast, is someone who is running a workshop, teaching others, and often innovating in their field. While people can go from journeyman to master studying under the same person who supervised their apprenticeship, it's not common. Studying under other people and working independently is seen as an important part of the process, a way for you to develop your own style and to broaden out from whatever your first teacher specialised in.
So even if Neville does go on to study more, he wouldn't be expected to go to New Zealand for it.
Neville grew up with a lot of emphasis placed on the family estate and legacy. He has a strong sense of duty to his family, and of the need to build it up and make it something he can pass on. So it's going to be a while before he does any formal sort of study again, but he does do the sort of scholarship lots of 19th century aristocrats did, and he does keep challenging himself with the gardens on the Longbottom estate.
I'll probably never write the story, but I had an idea for Neville ending up growing tea commercially - it started as a one-time gift to Augusta (of what turned out to be very, very strong black tea), and then it became popular among the Order of Hecate group, and then Hermione pointed out there would be commercial demand, and then Hannah Abbot ended up going into business with him as the managing partner (while he focused on the plant side of things), and of course the brand got a huge boost from being the tea that the Order of Hecate drink, even if a lot of people find it too strong. They drink it anyway, because it's fashionable.
Someday he'll probably end up getting his mastery the same way Molly Weasley did - with someone pointing out he's already working at that level.