NH resident, Employer in the state of Maine. I worked part time remotely in NH which has no income tax. Can I get a larger tax refund from Maine? If so how do I file ?

Genrally, the largest refund you can get from a state is the tax withheld payable to that state

@christopher-smal1 

Understood. However I was working part time in NH. I usually get a small Tax return for being a Non-resident to Maine because I live in NH working a full year. I think I have close to 6 months documented of working days of not physically working in Maine.  Wouldn't I be entitled to a potentially larger return ?

Yes, your refund will be based on how much of your wages were earned in Maine. 

If you look at your W-2 form, you will see your state information in boxes 15, 16 and 17. Based on what you are saying, I'd expect to see a line with Maine wages and another line with New Hampshire wages. The total of the two should add up to your total wages.

You will be taxed as a nonresident in Maine on the wages listed as Maine wages on your W-2 form.