Is there a word to call the second rainbow above the first?

Solution 1:

The specific one OP is talking about is called the secondary rainbow (as opposed to the primary).

There can be more than two. All except the primary are loosely called "supernumerary" rainbows, but supernumerary/stacker rainbow is also a specialised term for multiple bows appearing inside the primary, which are caused by lightwave interference rather than reflection by water droplets.

Solution 2:

It's usually called a double rainbow in the US.

Although I misunderstood the question originally (I did not realize that you were looking for a specific piece of a rainbow), apparently, the actual scientific term for the primary and secondary rainbows are just that, as explained by this guide.