Evolution doesn't apply to organic beings. It also applies to sentient ones, as well as abstract concepts like languages and cultures. Perhaps something similar could apply to our culture as well. We already established that they are hunter-gatherers, as well as decided their island, their taxonomy, and so forth. So how can we evolve it? Well, for me, personally, there are very few predictable things, so let's focus on those and worry about the rest at the right moment. Fair reminder, I'm new to this.
One thing is for sure, our hunter-gathers would be gathering more and more items, leading to a need to properly move too large amounts to other places. Perhaps they discover an idea somehow leading to the invention of the wheel, which, like the discovery of fire, occurred independently multiple times around the world. Perhaps from there, they could invent carts to carry the loads, and maybe transport groups of people that can't move at all. And paths would likely be marked before then, though there would eventually be more to mark, and the creation of wheeled carts and the utilization of local animals to move them would lead to new distances being achieved, and new settlements throughout the island.
Let's say that sometime after the creation of the wheel and carts, and the transition between writing systems, which will be discussed soon, yet before they start to spread outwards from their previously limited areas to occupy the entire island and the smaller offshore islands, the language finished reaching its evolved form. It'll split into four once they colonize the peninsulas and the smaller islands.
Marketplaces could be built, and maybe restaurants could be invented, both as people are offering goods like animal meats, edible vegetation, and interesting spices, seasoning, and juices. And perhaps inns could be established in case people need to rest before resuming their journeys to their destination. I'd envision a mixture of wooden houses and stone ones, with perfectly-structured designs to avoid causing problems. Well, unless a tsunami or a smaller level yet still dangerous wave ruins everything. Let's not think about that.
Perhaps these hunter-gatherers could also learn to establish a nation and build some temples for their religious practices, while also classifying their first city as the capital, and forming new cities along their island as they expand outwards. They might find other places that would be their new capital, just in case. We'll need a map of their island to figure it out. Anyway, afterward, sometime after their language earned some consonants that I thought of including since the very beginning. And it's about high time that we evolve the language.