If you had zero other income in the second state, you could be justified in claiming all of the interest income to your first state if for no other reason than it is easier to do so.  In addition, if the amount is small, it won't have any impact at all on your return.  Having all of the income from one state allows you to file as a full-year resident in that state even though you moved:  all of your income was produced while you were living there.

There is a caveat:  Technically, the correct procedure is to split the interest income proportionately between the two states.  Thus, if the interest income is considerable, to the point that you could have a tax liability in the second state if you report the portion of interest that was generated after moving there, then you will need to go through the exercise of filing both state part-year returns and allocating the correct amount of interest to the second state.  It's just not likely that you will have enough interest income for this to be necessary. 

As an addendum, my wife was shifted to part-time remote work for her company right when we moved. The compant is based in our original state and it looks like they never changed her address in her W2. Is there any reason to file in our second state?

Totally changes the scenario.  Yes, now you will need to file in the new state, because that's where you were living when the income was earned.  However, if the company was in DE, NJ, NY, PA, or NE, the original state still taxes the income.  Those 5 states tax the telecommuter who is working for a company within their state remotely.  Please feel free to comment and I'll provide additional information.

If it changes anything, technically, the remote work was done among several states. We left our first state in May, moved all our back accounts, car insurance (not registration though), and got health insurance at my parents house in another state. We then road tripped across 15 states and Canada where the work was done remotely. We didn't really"live" in the second state in a traditional sense.

Sometimes reasonableness must be applied.  If the first state was one of the 5 that I mentioned, treat all of your income taxable in the first state, and then also file a part-year return for the second state.   All of your income is taxable to state one, and the second state taxes the income you earned when you moved there, but will give you a credit for the tax you pay to state one on the telecommuting.  To prevent tax insanity, treat your income as being earned in the state where you moved to, as it would be impossible and impractical to file 15 nonresident state returns.

Now, if the first state was not one of the first 5, then file a part-year return for the first state, only including in income the amount you earned while living there (you may need to apportion or allocate your income), and then, for the second state, file a part-year return for the remaining income.