You do not need to include anything else. Do not send a copy of the 1099. Do mention it in the explanation; e.g. something like "increase tax exempt income due to corrected 1099-Int, no other changes". The explanation covers everything.
I would not bother with a amended return, since nothing changed. An increase in tax exempt income sometimes affects the Modified Adjusted Gross Income (MAGI) for the calculation of other tax attributes. The most common being how much of your Social security benefits is taxable.