yes it's too late. only 2017-2019 are available. 

NH does not have an income tax based on your wages, but there is a tax on interest or dividends,  If you had income from interest or dividends you can file a NH return to pay your tax due if you use a paper form and mail it in, but it is too late to use software for a 2016 return.   

Get the form from the state:

https://ttlc.intuit.com/community/state-taxes/help/how-do-i-contact-my-state-department-of-revenue/0...

Here is a case where the "Add a State" command is actually relevant.

Did you even try it?

You already have the fed part installed.

you would likely be charged.

@clemelin If you try downloading 2016 state software it will not be available---or even if it is, it cannot be updated so the results of using it would likely be inaccurate.   TurboTax no longer supports or updates 2016 software.