I don’t know where you are coming from and I’m certainly not a health insurance specialist. Two thoughts come to mind. Firstly, please please please make sure you have sufficient health insurance coverage in NY – getting sick in the US without health insurance is a huge financial risk and can easily bankrupt you. But you know that – that’s why you ask.
Since you will not be living in NY for a long time, often times travel health insurances are “good enough” – those insurances typically don’t cover anything that is foreseeable / not related to managing emergencies and getting you back home (e.g. they typically don’t cover dental work outside pain relief, pregnancies, chronic diseases etc.). But provided a travel insurance is valid for the US (check if that applies) and is not excluding professional activities (yes, your internship is not a full-time job, but it isn’t leisure travel exactly either, so better be covered) you should be all set and good.
I know that some countries have expat associations that offer such insurances. I know that even some credit cards have these kinds of insurances built-in. I also know of at least one big international health care provider (US owned) who has an international dependance in Beligum and who offer these kinds of insurances.