Polynesians didn't reach the eastern islands very long ago, estimates vary, but it would require an earlier settlement of Eastern Polynesia POD. It's not actually confirmed if Polynesians ever reached South America, it's entirely possible kūmara reached the Pacific Islands by sea. As for Polynesian influenced Taíno boats? I highly doubt Polynesian influence would be that big, or spread that far.
What would the Eastern Seaboard trading network even be for? Do the people there have anything the Taíno would even want?
Why would Taíno even start exploring the open ocean and going to places as far away as the Azores and even Bermuda? The ocean between the Americas and the Azores is devoid of islands.
A possible PoD--Eastern Polynesia is settled far earlier, so just a continual chain of migration across the Pacific and eventually to South America. A culture in the Galapagos and on Cocos Island facilitate the exchange of Polynesian technology, eventually reaching the Orinoco region where the alt-Taino expand and innovate on the technology, becoming expert sailors. But an indigenous South American development isn't unreasonable either.
What would they want to trade? A lot of stuff--long distance trade wasn't uncommon in the Americas. Textiles, plants (like yaupon), food, antlers, etc all make good trade goods. Especially if this trade network ("New World Phoenicians") is linked with Mesoamerica, then there's even more stuff to trade.
Why explore? Two words--Gulf Stream. This current sweeps past Bermuda and then to the Azores (which admittedly are a stretch). Going up the coast, the current carries past Newfoundland and to Iceland.
Two questions, what islands do you mean, and why would they need better boats? Their boats were fine for their lifestyle.
Jan Mayen, Bear Island, and Svalbard would be the main ones. All three could sustain a solid population of a few Paleo-Eskimo bands yet never did, thanks to the difficulty in reaching them through the lack of sea ice.
Also, did the Dorset or earlier cultures use umiaks or anything similar? By better boats, they need something which is seaworthy enough to cross the open North Atlantic, which you have to if you want to settle Iceland.