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After you file

Yes, amended returns can be e-filed for New York and Vermont state returns (not federal returnsonly. You will check the status of it on your state's website. And you can go here How do I track my New York tax refund? to get details on tracking state refunds for NY. 

It is also recommended to wait until your state has accepted, processed and you have received your state refund before you amend the return. Or if you owe them, you want to pay them before amending return. Do not make any changes in the return before then. 

You can find steps here I need to amend my state return for how to amend state return without amending the federal one. 

Can I e-file an amended state return?