Lucky you! The interior of Puerto Rico is quite mountainous and there's some great hiking to be done. Be sure to check out El Yunque, a sprawling rainforest—the only one on U.S. soil—that's maintained by the Forest Service.
Fiding the right footwear for Puerto Rico is difficult because it's so hot and humid. You'll want boots that are protective, but also light, breathable, and fast-drying. And you mention you have some fit issues as well, although there are ways to deal with those.
The Vasque Breeze ($160) is a good starting point. The Breeze is a nice light-to-midweight hiker with leather and mesh uppers that stay breathable in warm weather, a Gore-Tex insert to keep your feet somewhat dry, and nicely lugged outsoles for grip in soft soil or mud. It's a great boot for day hikes or light backpacking trips.
The Best Boots for Jungle Hiking: Merrell Moab Ventilator Mid
Merrell’s Moab Ventilator Mid ($100) are designed as a desert boot, with a focus on breathability, but they hold their own in damp and warm weather, too. Without a Gore-Tex liner, they're not the right boots for a downpour, but the lack of Gore-Tex will actually boost breathability in mixed conditions. And they’re extra lightweight—really a high-top running shoe—but are still comfortable and protective.
The Best Boots for Jungle Hiking: Lowa Zephyr GTX Mid
Lowa's Zephyr GTX Mid ($180) are similar to the Vasque Breezes, with a composite leather and nylon upper, a Gore-Tex liner, and an emphasis on breathability. I like their external polyurethane exoskeleton, which adds support without bulk. Like most boots in this class, they have nylon shanks for light weight and flexibility on day hikes or with a light load.
The fit in all of these boots will be improved with some Superfeet Green insoles ($40). They're the “high-volume” model, meaning they’re for low-volume and narrow feet.
My last piece of advice is this: Buy two pairs of boots and let one dry out while you're hiking in the other.